Approves of JH sharing HW's findings. HW on excellent terms with Augustus De Morgan. Suspects two integrals exist, one real and one imaginary, for P. S. Laplace's equation.
Approves of JH sharing HW's findings. HW on excellent terms with Augustus De Morgan. Suspects two integrals exist, one real and one imaginary, for P. S. Laplace's equation.
Testing recent work on equations analogous to P. S. Laplace's but of higher order. Solution so simple that HW expects to find it published elsewhere. Corrects error in previous letter regarding generality of arbitrary constants.
Demonstrates easier way to treat equations analogous to P. S. Laplace's.
Please correct error in HW's earlier calculations.
Corrects calculations relating to arbitrary constants in HW's previous letters. Encourages JH to avail himself of this information in second edition [of Examples of Finite Differences (1820)] .
Relative to previous letter [see HW's 1855-1-7], HW found two possible equations, due to ambiguity of signs.
Heard JH not well. Offers him 'a nosegay, gathered from [JH's] own garden of the Differences of the Powers of Zero.' More notes on the self-repeating series. Developed two equations that HW may submit to Cambridge [Philosophical] Society.
HW neglected to state principle from which he derived equation for eliminating differences. Expresses it in formula.
Glad that JH shows renewed interest in mathematics. Besides HW, JH, Augustus De Morgan, and Mr. Gerard, there appear to be no others interested in researching the differences of the powers of zero. Gave outline of HW's paper at Cambridge Philosophical Society. Plans to revise it according to suggestions from JH and De Morgan.
Quick way to determine coefficients in tables of differences of powers of zero.
Sends law of coefficients, for developing summation theorem.
Coefficients of differences of odd powers of zero observe same law as those of even powers. Example of table of coefficients of differences of 015. Table took 15 minutes to compute by HW's process of summation. Notes error in table for 012 sent earlier.
Compliments HW's expression for the formula for coefficients of differences of the odd powers of zero. JH busy, but hopes soon to develop demonstration.
Glad for JH's renewed interest in mathematics. At 72, HW expects to do no more. Reviewed JH's letter of 14 May [1856]. Fears JH misunderstood meaning of symbol HW employed. Quotes explanation from HW's paper on self-repeating series.
Reflects on relations between differences of same power of zero. Reviews procedure developed for determining terms and law of coefficients. JH's critique expanded role for these, but HW disagrees with JH's latest comments.