Dr. Stevelly on the Composition of Forces. 251 



proved that AD„ is the resultant of A B with BD n; by intro- 

 ducing with AB and BD„ the subtractive force represented by 

 D n x, and with AD H the subtractive force represented by D n B" 

 with B" x, we again see that AB with Bx must have the same 

 resultant as AB" with B" x. Hence AB' with B 1 x must have 

 the same resultant as AB" with B" x, and this can have no 

 other representative than A x without contradicting either what 

 has been already proved (in II.), or what has been now proved. 

 If, now, z coincide with x, it is proved that the resultant of A B 

 with B z is represented by A z. If not, z must lie either between 

 D ra _i and x or between x and D n . Now draw Ax' bisecting the 

 half angle within which z lies, and, proceeding exactly as before, 

 we show that Ax f is the resultant of AB with Bx 1 . If then, 

 again, z coincide with x', we have proved what we require. If 

 not, we again draw A x" bisecting the half angle within which 

 z is found ; and so ou without limit by continual bisection until 

 either z lie in the last drawn bisector, or it appears that A z is 

 the limit towards which the last bisector may approach closer than 

 by any angle however small. In the limit, then, Kz represents 

 both in direction and magnitude the resultant of the two forces 

 represented by A B with B z acting together on the material 

 point A*. 



* Since the above was printed, I have found that (2), (3), and (4) may- 

 be most materially shortened^and simplified, thus : — (2) and (3). Since in 

 (1) we have seen that ADi represents both in direction and magnitude the 

 equivalent or resultant of the two equal forces represented by AB and BDi, 

 if with each of these we introduce a force acting at A represented by Di D2, 

 we shall see that the resultant of P acting with Q, now represented by B I> 2 , 

 must be the same as that of the two equal forces represented by A Dj and 

 Di D 2 acting together at A. But since A D^ bisects the angle Di A C, its 

 direction must be that of the resultant of the two latter (by Cor. II.), and 

 therefore of the resultant of P and Q represented by A B and B D 2 ; and 

 (by Laplace's Principle III.) the magnitude of that resultant is represented 

 by the length of A D 2 ; A D 2 then in this case represents both in direction 

 and magnitude the resultant of A B and B D2. Similarly, by introducing 

 along with each of these at A another force coinciding in direction with Q, 

 represented by D 2 D 3 , the new resultant of P with Q now represented by 

 A B and A D s is shown to be represented in direction and magnitude by 

 A 1) 3 . And so on for AD 4 . . . A D n , n being any whole number however large. 



Again, (4) suppose, we have now proved that the resultant of P and Q 

 represented respectively by AB and BD n -i is represented by AD M -i 

 both in direction and magnitude, and that the resultant of A B and B T> n -i 

 is A D n -i, and that of A B with B D w is A D M . Let A x, as above, bisect 

 Dn-i ADw, and introduce with the first pair and their resultant a force at 

 A represented by D n -i X, then the resultant of A B with A x will be the same 

 force as the resultant of AD»_ 1 with D n -i oc. Also, by introducing with 

 the latter pair (AB with BD«) and their resultant (AD W ) the subtractive 

 force at A represented by Dn x, we shall again have the resultant of A B with 

 B#the same as that of AD W with D n x ; therefore this resultant must be the 



