44 Capt. A. Cunningliam on [May 11^ 



_^-( y^4-l)(^^H-2).. ..(7. + n -l) 



+ 12 |n-2 --^ +11 \n-l ^ + |r ^ * * 



(1) 



(2) 



They wall be distinguished as of nth order, rth ra^A', and Ath class ; r 

 and Qi are supposed always positive integers, and r not >ii; and h may 

 be any quantity whatever. It is obvious tbat there are 



(n+1) Clairautians of nth order ; ^ 

 (r + 1) terms in a Clairautian of rth rank in general ; 

 {l—h) terms in a Clairautian of rth rank, when Tc is zero or a 

 negative integer numerically < r. 



Thus a Clairautian is a differential expression, the order and rank of 

 which determine the orders of its highest and lowest differential co- 

 efficients ; it is also obvious that the difference of the exponent {jp) of 

 X and order {n—r-^p) of differential coefficient is the same in every 

 term. 



2. Clairautian Equations. — It is proposed to term a differential equa- 

 tion involving Clairautian functions a Clairautian equation. Upon 

 the important properties proved in arts. 4, 5, the solution of many such 

 differential equations may be founded and effected with elegance. These 

 will be developed in what follows. 



In consequence of the limitation of r, n as positive integers (art. 1), 

 the differential equations presented will all be of the ordinary type, that 

 is, involving the differential symbol D only in a rational integral form, 

 " General differentiation " will be freely used when necessary to the 

 generality of a solution (so that the quantity Ic may have any value). 

 Any difficulty that may be felt in the interpretation of the transcendent 

 D*0 (when Tc is not an integer) will generally disappear in the final re- 

 sults, such transcendents, in fact, cancelling. 



3. Algebraic relations. — It is easy to see that the Clairautians of zero 

 rank (r = 0) are simple differential coefficients. 



^Uo.o = 2//-II,,, = 2/',^lIo> = 2/", ^■Uo,n=y"^ .... (3.) 



