1884.] identified with Sir J. Cockle's Criticoids. 51 



These relations are given by Sir James Cockle,* and since x y = (j^) '» 



they are obviously the same as Professor Malet's forms Ij and <I 2 , 

 viz. : — 



Q3' + 3Q 1 Q 3 _F8H3P 1 P 3 (I 



Qi +2Q/-3Q, _ V l ' + 2?* -3P t 

 Q3I P? 



see Malet, p. 768. 



12. These forms are not, as in the case of the first class of criticoids, 

 independent of the order of the quantoids. This appears from some 

 general results obtained by Sir James Cocklef by the following pro- 

 cess. Let t and x be connected by the relation 



dt 



[dt 



!= Jt 



where T is a function of t; then y^j ~~( T-j^J , which gives, by 

 symbolical development, 



A 8 " 1 



¥\dx) \dt) 2 T * \dt ) 



m 



n.n-1. n-2 /T" Sn - 5 r 2N 

 2.3 \T 4~'T^ 



-f&c, 



where differentiations with respect to t are denoted, as before, by 

 grave accents. This formula enables us to change the independent 



variable in any given quantoid in J^, from x to t. Thus the general 

 linear differential equation 



(i > p 1 ,p a> ...P,j^i) i V=tf 



may be transformed into 



(i.qlq*. . .Q,rjj-,iyv=o, 



* Cockle. u On Linear Differential Equations of the Third Order," " Quarterly 

 Journal of Mathematics," vol. vii, pp. 316-326, more particularly equations (/) 

 and (0) on p. 318. This paper bears date 15th August, 1864. See also in the same 

 Journal, vol. viii, pp. 373-383, and vol. xiv, pp. 340-353. 



f " Ona Differential Criticoid," " Philosophical Magazine " for December, 1S75, 

 vol. 50, ser. iv, pp. 440-446. 



E 2 



