Sir James Cockle on Primary Forms, 349 



23. Put l + x 2 = — t 2 , and suppose (12) to be given under 

 the form 



y" + (L,M,N)y=0, (27) 



wherein 



(L, M, N) = ^- 2 r 4 (L + M^ + m> 4 ), 



and L, M, and N are, or are taken to be, one- valued. 



24. A change of the independent variable from x to x" 1 

 changes t 2 into x~H 2 , and (27) into 



/'+V + (F,M,L)y=0; 



and a change of the dependent variable from y to .a? -1 ?/ changes 

 the last equation to 



y + (N,M,L)y=0; (28) 



so that we have interchanged L and N. 



25. The expression 



L + M^ + N^ 4 



being identical, save in form, with 



l-^2-(M-2N)« 3 + N« 4 , 

 the change from x to t changes (27) into 

 d 2 y dt d 2 x dy m dx 2 __ 



dtp az U7i 7u ">" -*-tt*2 y—v, 



wherein 

 but 



T=^- 2 r 4 {l-^ + (2N-M> 2 + ^ 4 } 

 dx t d 2 x 1 1 





dt x dt 2 xl + t 2 



consequently we have 



d 2 y 1 dy 



dt 2 til + ^'dt^ 



{l-M 2 + (2N-M> 2 + m i \t-%l + t 2 )-*y; 



and if we change from y to fi(l-\-t 2 )~^y, and then replace t 

 by x, we shall have (27) transformed into 



y " + (l-M>, 2N-M-|, N)y=0. . . (29) 

 26. Denote these transformations by/ and cf> respectively, 

 and let V mean (L, M, N) ; then we have 

 V=(L,M,N), 

 /V=(N,M,L), 

 4>~V = {\-M\ 2N-M-|, N), 

 /<£V=(N, 2N-M-|, i-iE 2 ), 

 */V=(i-^J, 2L-M-f, L), 

 /*/V=(L, 2L-M-I, i-2E*). 



