Cambridge Philosophical Society. 453 



From these are obtained results in complete analogy •with those 

 for primordinal equations. But when J'ai/'A)— >/'A>i^aj^=0. the usual 

 criterion of singular solution, is made valid by \^a=0, ;|/4=0, a sin- 

 gular primitive of the singular primordinal may be obtained, which 

 does not necessarily satisfy y"=x- 



10. Similar forms are given for ^riorrfma/ equations. In noticing 

 the manner in which the equations of the general theory may be 

 easily expressed by what are called determinants, Mr. De Morgan 

 expresses his admiration of the system, and his sense of the important 

 ser^'ices rendered by those who have laid its foundations. But he 

 refuses to employ the word determinant in the sense proposed, on 

 account of its not expressing any distinctive property of these func- 

 tions. Until those who have a better right to give a name provide 

 themselves with a distinctive one, he intends to call them eliminants. 



The forms connected with y = \p(x, a, b) may be easily translated 

 into others derived from (j)(.v,y, a, b)=0. But the formula which 

 connects x>/' with (p is as follows : — 



V ,= (log/'M*£!ldM^2'\l -j- 'P'''t>iy—4>i><t><'!/ ^ 



^ \ \ <p^ ) i Ky <P'a't>My—'l'bi>(a:\ff 



•where by U^i^ is meant Uj:+U^y', even when U is a function of y'. 

 Thus (xy'—i/)x\,/ , as here used, is 0. 



1 1 . The following idea of reciprocal polarity has been presented 

 by M. DruckenmiiUer (as cited from Crelle's Journal by Mr. Boole), 

 and, independently, by Professor Boole : it occurred to the author of 

 this paper before he had seen the researches of either. If there be 

 equations involving m-\-n variables, and if, determining a point by 

 fixing m of the variables, a curve be determined by giving all possible 

 ■values to the remaining n (point and curve being here merely names 

 of objects determined), we may say that the (?«)-point is the pole of 

 the («)-curve. Similarly, we may make each (?i)-point the pole of 

 an (TO)-curve. And all the points of any curve have polar curves 

 which contain the pole of that curve. If the two sets of variables 

 be severally made primordinally compensative, the general properties 

 which arise are easy extensions of the well-known theory of reci- 

 procal polars. Let (x, y) and («, b) be two points : the polar pro- 

 perty of x2+y-=ax-f-6y contains the direct and converse property 

 of the angle in a semicircle. If (p{x, y, a, b) be the modular equation, 

 and if x, y and a, b be compensative, any element (x, y, y') of any 

 (x, y)-curve to the pole (a, b) determines an element (a, b, b') of an 

 (fl, 6)-cur\'e to the pole (o", y). These curves are reciprocal polars. 

 In the common system, the modular equation is linear with respect 

 to both pairs of coordinates, and the locus of those poles which lie 

 in their polar straight lines is a conic section, to which the polars 

 are tangents. 



12. The method of transforming differential equations, given by 

 the author in his last paper, is precisely the reference of the curves 

 sought to their reciprocal polars, the modular equation being taken 

 at pleasure. Mr. De Morgan now proposes to call it the method of 

 polar transformation. Let ip(x,y, a, 6)=0 be the modular equation. 



