294 Mr DE morgan, ON THE GENERAL PRINCIPLES OF WHICH THE 



This equation, difFerentiated twice with respect to ,v, and twice with respect to y, shows that 

 d>"ai : <bx and (b"y : (by are equal, independently of all relation between x and y. Hence 

 (px is ^e'"" + i^e"""^, and substitution in the equation shows that A = B = ^, and that m is 

 arbitrary. Hence, as Poisson has done, we easily show that (pa? = cos a?. 



Or thus: — decompose one P and the aggregate on the other P and on its perpendicular: 

 we easily obtain 



2 ((pyY = 1 + <^ (Sy), 202/ ^^ - yj = ,^ ^^ - 2y j . 



Each of these equations, taken singly, has an infinite number of solutions : taken together, 

 there is no common solution but (py = cos ay. 



The preceding demonstration does not assume that tendencies are applied at one point, nor 

 even in one right line, but only that they have definite directions in a given plane. If, 

 howevei", any full definition of tendency demand the idea of a point of application, it follows 

 that the line of application of the aggregate passes through the intersection of the lines of 

 application of the aggregants. For if not, A^B being C, {-A) ^ (-B), by 2, would be C, in 

 a line related to the angle of —A and —B in the same manner as C to the angle of A and B. 

 But C and C' counteract, and application at one point is now essential to counteraction, as well 

 .as opposition of direction. Consequently, C and C' are in one line: and there is no one 

 line related similarly to both the angles of A and B, and of — ^ and -B, unless it be a 

 line passing through the point of intersection of the lines of A and B. 



2. When different applications mean different points to which one direction is applied, the 

 aggregate is the algebraical sum of the aggregants, applied at a point in the line of the 

 applications of the aggregants, distant from these points of application inversely as the 

 aggregants, internally or externally, according as the aggregants are in the same or opposite 

 sides of their common direction, or, as it might be said, in the same or opposite suhdirections. 



The point of application of the aggregate must be in the line which contains the points of 

 application of the aggregants. For if A^B be on one side of the line of A and B, then by 

 postulate 2 and a half revolution of the plane about that line, - A and -B have an aggregate 

 - {A^B) on the other side; whence, by 6, A and B have also an aggregate on the other side, 

 by which l is contradicted. The same sort of proof shows that the aggregate must be 

 parallel to the aggregants. 



Again, the aggregate of A and B is in magnitude the sum of the aggregants. Let A^B 

 be C, the actual distance of the points of application being c, a concrete magnitude. Hence, 

 as proved, C : A depends only on c and B : A, and as the magnitude c cannot be expressed* 



• The celebrated controversy which arose out of Legendre's 

 theory of parallels was conducted (to the best of my remem- 

 brance) without any reference to the following point, or at least 

 without any clear use of it. In every case but one, it is impos- 

 sible to conceive number a function of magnitude : that is, one 



concrete magnitude being given, and no other thought of, the 

 actual determination of number cannot follow. There is no 

 number which can necessarily be brought before us as a conse- 

 quence of the English foot being what it is, without reference 

 to any other length, or the same length repeated. But a number 



