Sir James Cockle on Primary Forms. 141 



which, though of the soluble form of art. 6, is regular. It does 

 not apply to 



d q y cot x dy 2 _~ 



~aV*+~3 dx + 9 y ~°' 

 which is a coresolvent. 



13. But the reciprocity of forms will appear if in (12) we 

 change x into x\/ — l, thus obtaining 



(l~^) 2 ^^- + (L-M^ + N^ 4 )2/ = 0. . . (24) 



Now the last coefficient may be written 



L(l-^) + (L + N-M> 2 + N^ 2 (^~l), 

 so that (24) may be written 



dx* + \x*(l-x 2 ) (1-a? 2 ) 2 l-x*P ( -' 



Changes into sin.2?, then (25) becomes 



g + tan*f +Qj,=0, .... (26) 



where 



sir« cos^a? 



Taking the criticoid of (26), we obtain 



where, after due substitutions, 



R = L-N-^] 2 + i + Lcot 2 ^-(^E 2 -i)tan 2 a7 = Lcot 2 «r 

 -f m + 1 + n tan 2 x, suppose. 

 The last differential equation will be solved regularly if A be an 

 integer, and if iE 2 — J =,/(,/ + 1), where j is an integer, i. e. if 

 M is half an odd integer. Hence a primary form will have 

 become regular. Again, let 



« 9 - = m-L-/i + l=i-N = (E + i) 2 ; 



then, if E be an integer, 2« will be half an odd integer, and, A 

 being entire, a regular form will have become primary. 



14. If in (25) we change x into sec x and take the criticoid, 

 we obtain a result in which the above value of R is replaced by 



R = N-L-^E 2 + i-(^ 2 -i)cot 2 ^ + Ntan 2 ^ = Xcot 2 ,r 



-f fjb + 1 4- N tan 2 x, suppose. 

 Here 



^ =/ ,_ X -N + l=i-L=(A-i) 2 , 



and corresponding inferences may be drawn. 



