TBANSACTIONS OF SECTION A. 605 



Society,' No. 235, 1884. The case when the dependent variable is changed is 

 first considered. Starting fi-om the two linear difterential equations of the w-th 

 order 



(l,P„P„...P„)(|,l)V = 0. . . . (1) 



(l,Q„Q,„ . . •Q'.Jd^.-iy'—O • • . • (2) 



in which the dependent variahles are supposed to he connected hy the relation : — 



logy = logs+y(Q,-PJ(^x . . . . (3) 



and introducing a third dependent variable (u), the author obtains two other 

 linear differential equations of the n-th order, viz. — 



(i,R„p., . . . R«)(l.'0"'^ = ^- • • • W 



(l, S„S„ . . . S„)(|,l). = 0. ... (5) 



(4) being connected with (1), and (5) with (2) by the respective equations 



log7/ = log 2;-/Pjf/.f (6) 



logs = log t;-y*Qjrf.f (7) 



Equations (4) and (5) obviously become identical when 



R, = S„ R, = So, . . . R„ = S„ . . . . (8) 



and this system is necessary and sufficient to determine the relations of the 

 functions Pj, P,, . . . P„ and Qj, Q^ . . . Q„_ so that (1) and (2) may be con- 

 nected by (3). The author calculates the criticoidal forms given by the system (8) 

 as far as Rg = Sg, and he obtains results which are all included in the formula 



^^(p)-'£;?.-^.(Q)-'^ • . . . (0) 



r denoting the degree of the criticoid. In particular he finds 



^,(P)=P,-P,^ (10) 



^3(P)=P3-3P,P, + 2P,3 (11) 



(9, (P)=P4-4P,P3-3P,2 + 12Pj»P2-6P,* . . . .(12) 

 6, (P) = P5 - 5PiP^ - 10P,P3 + 2OP/P3 



-60Pi3P„ + 30P,P.2 + 24Pi^ (13) 



e, (P) = Pg - 6P,P, + 30P>P, - 15P,P, 



-120P,3P3 + 120P,P2P3 - lOPg'^ 



+ 360Pi*P„ + 30P/-270P,'^P2,-120P,« . . . (14) 



Of these results the first three, (10), (11), and (12), agree with those already 

 obtained by Sir James Cockle and Mr. Harley, and the last two, (13) and (14), 

 are now published for the first time. The advantage of the method here employed 

 is that the system (8) gives at once R„ = S„, where R is a function of P, and S the 

 same function of Q, whereas by using (1), (2), (3) we are led to Q„= a certain 

 function of P, and have to obtain the criticoids by means of elimination and other 

 contrivances. A similar remark applies to the case of the change of the inde- 

 pendent variable next considered. 



Let (l,<^i(^),</>.(-t') . • • <^„(-v))(§''l)'V = . . . (15) 



(l,V',(.r),V',(>), . . . >/'„(.v))(|,l)'V = . . . (16) 



