or the Incriticoid of the Fourth Degree. 457 



The object of this paper is not to expound the general 

 theory, or to point out any of its numerous applications, but 

 simply to place on record the explicit form of the quartine or 

 incriticoid of the fourth degree. This form I calculated in 

 February 1885, and communicated at the time to Sir James 

 Cockle ; it has not hitherto been printed. Unique in cha- 

 racter, its peculiarities deserve attentive study. I hope to 

 consider these in a memoir dealing with the general subject. 



Using the quantical notation we may write the linear dif- 

 ferential equation of the nth order thus — 



(i, p„ p„ . . . p„x^, i)» y =o, 



where P x , P 2 , . . . P n are functions of .r. Changing the inde- 

 pendent variable, this equation may be transformed into 



(1, Q,, q 2 , . . . q„x|, i)"y=o, 



where Q 1 , Q 2 , . . . Q„ are functions of t. Denoting differentia- 

 tions with respect to x by acute, and with respect to t by 

 grave accents, and representing the general quartine by 



»(Q, Q', Q", Q"') 



4 ' 



or its equivalent 



4>(P, F, F', P") 



P 5 



the result to which I have been led, omitting here all details 

 of calculation, may be exhibited in the following form, 

 viz. : — 



+(Q, Q', Q", Q"') =Q1" +~ Ql' Q.h-^1 (Q\)> 

 + ( ^QiQ?-i^+ioQ3Q l+ f^Qi 



3(n-2) n n2 , 15n 3 -7 5 n 2 + 1 20n-38 n4 



