On the Inverse Problem of Criticoids. 18$ 



The question of nomenclature was avowedly introduced to 

 show that Dr. Mills had involved the subject in some obscu- 

 rity, and that therefore, if we had misunderstood him in the 

 more important matter to follow, we must ask for indulgence. 



The remainder of our paper was occupied with a discussion 

 as to the accuracy of his deductions from his equations. He 

 himself says (loc. cit. p. 574) it is important to illustrate these; 

 and he can therefore hardly stigmatize as triviality the ques- 

 tion as to whether the illustrations are correct. 



XXIII. Inverse Problem of Criticoids. By Sir James Cockle, 

 M.A., F.R.S., F.R.A.S., F.C.P.S., Mem. Lond. Math. Soc, 

 Corr. Mem. Lit. and Phil. Soc. Manchester, Hon. Mem. 

 Roy. Soc. New South Wales, late Pres. Queensland Phil. Soc* 



1. TN a " Supplement on Binomial Biordinals," recently 

 J- printed in the { Proceedings of the London Mathema- 

 tical Society 3 (vol. xiii. pp. 63-72), I have, in certain cases, 

 linked a binomial of the second with one of the third order 

 wherein the symbolical factors of both numerator and denomi- 

 nator are in arithmetical progression. The theory of criticoids 

 expounded in these pages sheds a light upon this result and 

 gives a foreshadowing of others. 



2. All binomial ter ordinals may be included in 



d?y S a + ex n d 2 y 3 f+gx n dy 1 h + kx n 

 dx z + a l + x n clx 2 + x 2 l + x n dx + a* l + x n ^ 



3. For brevity I put 



l + x n } a -{-ex = X, %, 



/+ gx n , h + kx n = a 2 , a 3j 



and denote differentiations by accents, thus, and by a multi- 

 plication, changing the ter ordinal into 



Xx 3 y r// + Za^x 2 y" + Sa 2 xy f + a 3 y = 0. 



4. Take the criticoid, viz. deprive the equation of its second 



term by assuming a new dependent variables, =yx n X\~~n~) } and 

 divide by the leading coefficient. We thus get 



wherein s and t are defined by 



XVs = L + M^ + Kr 2n , 

 XV* = P + Qx n + R» a " + S.t- 3 '\ 



* Communicated by the Author. 

 Phil. Mag. S. 5. Vol. 12. No. 74. Sept. 1881. P 



