516 Mr DE MORGAN, ON SOME POINTS 



1. Further consideration has confirmed and extended some views on differential equa- 

 tions published in Vol. IX. Part n. : it has also indicated points of connexion between 

 subjects there treated apart. Before entering on the subject, I dispose of some preliminary 

 matters. 



It is desirable to invent single words to stand for the phrases "of the first order," 

 " of the second order," &c. I propose primordinal, biordinal, triordinal, &c. The word 

 differential may be dispensed with, since these adjectives are understood to apply to differ- 

 ential equations only. 



The symbol U x , which I use for the partial differential coefficient of U with respect to m 

 as explicitly contained in U, may be extended as follows. The complex coefficient, with 

 reference to x directly, and to p, q, r, as functions of x contained in U, may be denoted by 



u *\p, q , r , which thus stands for U * + U pPs+ u S*+ U r T x- The symbol U xUiPtq>r might be more 

 correct : but it very seldom happens that we want to exclude the explicit effect of a variable. 

 Parentheses may, when required, be employed to separate all that relates to one differentia- 

 tion : thus U {x \ p , Q ,r)(„iu,v,w) denotes d?U : dxdy, on the supposition that p, q, r, vary with x, 

 and u, v, w, with y. Thus, if p, q, r, be not functions of y, nor u, v, w, of x, we have 



xy \P* Q* r, u, v, w {x\p,q,r){y\u,v t w)' 



When the suffix notation is inconvenient, D may indicate differentiation, the specification 

 being within brackets: as in 



D*[xy\p, q, r, u, v, w]U = D 2 [(x\p, q, r)(y \ u, v, w)] U. 



When repeated specification is useless, we may preserve an indication of complexity by 

 using U„ 9 U mV or U (lUW) , as required : or we may, with due warning, omit all symbols of 

 complexity in any particular case. 



Differentiations may be expressed as in 



d x U= U x dx, d Xty>z U= UJx + U y dy + U„d%. 



A relation, the character of which there is no occasion to distinguish, may be expressed 

 as in (x, y, «) = 0, (a, b, e) = 0. A letter may be used for its own functional symbol : thus 

 u = u (<v, y, x) may denote that u is a function of x, y, x. By the phrase " for « write 

 u (x, y, #)" we may signify that we substitute for u its value in terms of x, y, z. Remem- 

 brance of distinctive functional letters, when the only idea needed is that of unspecified 

 functional relation, is often a heavy tax on the reader's thought. 



2. When a variable so enters that the partial differential coefficient vanishes, so that the 

 same primordinal forms of relation exist which would have existed if that variable had been 

 constant, it may be termed self-compensating : as in <p (x, y, a) = 0, <p a (x, y, a) = 0. When 

 two or more variables are so related to others, as in <p (x, y, a, b) = 0, <p a da + <p b db = 0, they 

 may be termed compensating, or compensative : all primordinally. When also 



<Pa^») da + <pH*\v) db = °> 



they may be called biordinally compensative : and so on. The whole system is always 



