Faraday to Jacob Herbert   25 October 1845

Royal Institution | 25 Octr. 1845

My dear Sir

Our recent conversation makes me think it desirable to write to you in reference to the subject & matter of my last report¹. First, as to the times of taking the observations from which the proportion of light to the oil consumed is deduced. These were selected from the middle of the periods during which any change was likely to happen in the lamps thus they were not made immediately upon lighting the lamps but an hour afterwards & at intervals of 1 or 2 hours & if the Argands had fallen & had to be retrimmed no observation within perhaps an hour of such change was taken[.] Every precaution that I could think of to give a correct mean average result was employed. The observations extended through seven days of from 12 to 14 hours per day & the lamps were burning all these hours[.]

Next I will speak of the light produced, its disposition & consequently its intensity in any given spot i.e. independant of irregularities in the form of the reflectors &c for these make a series of accidental changes in the particular distribution & intensity of the light that cannot be anticipated.

The rape oil lamp gives an average light which is to that of the Sperm oil lamp as 1 1/2 to 1 (see report of 9th instant) and the consequence is that in open space the former will send out 1 1/2 times as much light as the latter to an observer wherever he may be placed. When the lamps are fixed in reflectors the action of the latter is to collect a great proportion of this light & send it in a certain direction which in the Trinity arrangement is as nearly as is practical horizontal diverging from that direction only to a degree determined by the size & form of the flame.

The opening of the aperture of the Parabolic reflector viewed from its focus is about 120˚ or 130˚ and therefore we may say generally that such a reflector with a lamp in its focus collects three fourths of the light of the lamp, for it encloses about three fourths of the rays issuing from it. The proportion extinguished makes no difference in the consideration of the results[.]

If in the case of the rape oil lamp & the Argand lamp these three fourths were reflected in beams equal in form & extent of course the intensity of light in these beams would be as 1 1/2 to 1 or if not having equal forms their extent i.e their sectional area were equal then also the intensities of the lights might be considered as 1 1/2 to 1 and in any case the whole of the light in the areas they would respectively illuminate would be as one & a half to one. But it would not follow upon that that the intensity in any point of the one area to any point in the other area would be as 1 1/2 to 1 because the first area might be twice the size of the second & then the intensity instead of being as 1 1/2 would only be as 3/4 to 1.

To apply this reasoning generally to the two lamps when in the reflectors we have only to consider the areas into which the original portions of light equal to 1 1/2 and 1 are collected. These are determined by the divergence and the divergence is determined by the size & form of the flame[.]

The rape oil lamp has on the average a vertical divergence of 19˚.66 and a horizontal divergence of 12˚.5 and these multiplied together will sufficiently well express the area illuminated by it which is nearly 246 parts. The Sperm oil lamp has an average vertical divergence of 16˚.25 and a horizontal divergence of 16˚.5 which multiplied together produce an area of 268 parts, so that within these respective areas the light of the rape oil lamp is even in a higher proportion than 1 1/2 to 1 because it is more concentrated.

But the shapes of these beams are not perhaps those best fitted to illuminate the spaces required to be illuminated by a light house for though it may cast its light beneficially to a great extent horizontally & is required often to do so for uniformity of light in that direction still to do the same beyond a certain degree in a vertical direction is more or less to waste it and indeed to undo the use of the reflector[.] Let me therefore view the two reflected lamp lights under this aspect.

The horizontal divergence of the Sperm oil flame is 16˚.5. If by a properly shaped reflector (as suggested in the report) we were to increase the horizontal divergence of the Rape oil lamp beam from 12˚.5 to the same amount or 16˚.5 we should increase its area to 324 parts and of course diminish the intensity of the light in the same proportion and now that we have equal horizontal divergence for the two beams we can perhaps better compare the intensity of light for lighthouse purposes. This in the Sperm oil lamp 1 of light is spread over an area of 268 parts: in the Rape oil lamp 1 1/2 of light is spread over an area of 324 parts or 1 of light over 216 parts: hence the former light is more dilute & the latter more intense according to these numbers 268 and 216 or as 1.24 to 1.

So then with equal horizontal divergences the Rape oil flame gives a more intense light than the sperm oil flame in the ratio of 1 1/4 to 1 nearly.

Supposing that a vertical divergence of 15˚ or 16˚ is all that is required or is useful I have already shewn in the report referred to that we are unable to bring that light from the rape oil lamp which passes outside these limits into them. Now though this be true theoretically we must not forget that practically the beam is not equally diffused over the 15˚ or the 20˚ of divergence but that the center part is usually much the most powerful so that though we cannot collect the whole light of 20˚ vertical, in 15˚ we can select the central 15˚ and therefore greatly improve the illumination over them above the aforesaid proportion of 1 1/4 though we cannot raise it (within such limits of space) up to 1 1/2[.]

Excuse the liberty I have taken and the simplicity with which I have endeavoured to state these principles & points²

and believe me to be My dear Sir | Ever Truly Yours | M. Faraday

Jacob Herbert Esq | Secretary | &c &c &c | Trinity House