820 Cambridge Philosophical Society. 



March 10. — A paper was read by L. Barrett, Esq., on the Distri- 

 bution of the MoUusca on ,^. Qqs^ti g^^Jto)«fiy. , X W«.Axmal» of 

 Nat. Hist. May 1856.) ^^^ v^,^^^yi ,m V '"^V^V^ ^^ ^"^ 



April 28. — Also, a paper was read " On the Question what is the 

 Solution of a Differential Equation." By Professor De Morgan. 



This paper is a short supplement to § 3 of a paper on some points 

 of the integral calculus (Camb. Trans, vol. ix. part 2). It discusses 

 the principles on which such an equation as y'*=a-, giving 



is generally affirmed to be completely solved when br&e. It-dwells 

 on the distinction between a relation and an equation, which may 

 express the alternative of one or more relations ; it points out several 

 cases in which conclusions applicable to the simple relation only are 

 affirmed of any equation ; and, with reference to the question asked 

 in the title, discusses the manner in which the answer depends on 

 the cross-question, what degree of discontinuity is allowed to be iniK. 

 plied in the word 5o/m?ww ? v^ 



May 12. — A paper was read by Mn WaiAurVon oA Self-repeating 

 Series, in continuation of a former pap6r. 



The author showed in his former paper on self-repeating seriea<- 

 printed in vol. ix. part 4. of the ' Transactions ' of the Society, that 

 in the fraction which generates a series of either of the following 



^^' . V . [" < ^^j^^^^i, iBoidqoaoiiilS 9riJ rrj 



dou- on bib I "hifi^mM Hii^t^mc>llV3fipi(i &i imiiuo omfivli??, 

 ^tva \j«ft Yd b^ouboiq si doidvy^ ined * adJ isdi ,?.i3W wj?I xlA .^nrri) 

 aiii doiJv/ ?.all««+'^ + 2^'»+^^-^'S^*»+^r4- &c/:-:^i°^ iBnohioqoiq ei 

 P„.,, ui^rxv^ ' . T "* t'^* "^ bsavloijoofo 



the numerator of such fraction is a recurrent function 6f/. He also 

 then determined the coefficients of the several powers of t in such 

 numerator to be given linear functions .of the; differences (as the case 

 may be) of 0^", or of 0^'*+^ ^- -^^ '^•t'r ii 



In his present paper, from the n pairs of equal coefficients which the 

 recurrent numerator contains, the author obtains n linear equations 

 between the 2n differences concerned ; and selecting any n of these 

 differences, he concludes that each of them can be expressed in terms 

 of the other n differences not so selected ; and consequently that no 

 formula, expressed in terms of the differences of 0^ or 0^"^^ need 

 contain more than n of those differenccil i^'Wb) - I'J^ 



He gives the equations requisite for'obtluiiing A**'^''(0'^') in terras 

 of (A*^ A""\ . . . A^ 4')02'*; and ^«+i+/'(o2»+i), in terras of 



(Afffi i>Jl^^^a^^3i^A?J<;a?)0''*"*''* ; and he applies these and other of his 

 equations to the elimination of particular differences of zero fr^lf^q 

 sundry formulas. ,., 



A|£|^^,Mr. Bashforth exhibited models illustrating the Moon's 

 moabn. 'r 



