Integration of Differential Equations. 277 



dy cf-y . dy 



(m— l)r'"~2 __j_^2 -_ a;2»n-2^ or since M=a;™, we get ^ = 



mil dy A [dTi ^r -w u dy , m-u^ d'^y , , . 



f-, and \»^/ =m(m-l) -^+ ^ ; by subsntu- 



X du ^^ x^ du x^ du^ 



dy \dxl ** 



ting these values of -r-, , and x^=~ in {h) we get by a 



,. , , d-y I A — 1\ dy B^ y 



slight reduction ^^ + iH-^^ j ,-^7.-,"^ ^=0' (^)' which is 



A-1 



reduced to the form of (2) by putting 1-| =2pq—q-\-l—c, 



B2 A-1 



^^^^' 2g'-2=-l, or q = ^] .■.p = i + -^, also a^ = 1 : 



hence the integral of (k) is found the same way as that of (2) 

 given above ; hence we have y expressed by a known function 

 of 21, then putting for u its value x"", we have y expressed by a 

 known function of x as required. Again, if we put ^=6"', we 



dl^-^] 

 dy dydu dy dy \dxl d'y 



S"^ dx'^duTx=''d{i' ='''' du' ^""^ -d^^''~'' d^' + "* " 



dy 



^, substituting these values and M=e"* in (i), we have by a 



d^y I A\ dy B^ y 

 small reduction ^+^l + -j "^-^^ ^=0, (Z)j hence if we 



A B^ 



put2p5r-gr_i_i = i4.-, a2 = i, 62 = _^2j-2=-lor^ = i, we 



shall havep=J+— ; then the value of y is found in terms of u, 



as in integrating (2), and by putting for zf its value e"'', y be- 

 comes known in terms of x as required. The equations {h) and 

 {i) were proposed in the Mathematical Miscellany by Prof. Peirce, 

 at p. 399, first volume ; we gave an answer to them in No. 8 of that 

 work, which was incorrect in several particulars, which we shall 

 not stop to notice any further than to observe, that m, the inde- 

 pendent variable, when integrating with reference to x in La- 

 croix's integral, and v of our own, was involved in one of the 

 limits of the integrals, so as to be a function of x or v, which ought 

 not to have been so, but the error was not noticed by us in time 

 sufficient for correction before the solution was published. 



