of the Product of two Functions. Ill 



= twice the area of the figure enclosed by the third curve 

 traced out by S 3 , and by the lines = «i, = * 2 . 



The arrangements for describing the third curve mechani- 

 cally are obvious. An arm, OS^, capable of turning freely 

 in the plane of the paper round an axis through 0, bears two 

 collars, S x and S 2 , free to move along OSiS 2 ; and to these, 

 two equal arms, SiC, S 2 C, are freely jointed, being at the same 

 time each free to turn' in the plane of the paper round an axis 

 through C ; the two collars bear pointers with which to follow 

 the two curves. A second long arm, OT, can turn round the 

 axis through ; at T is a collar free to move along OT, and 

 to it is rigidly attached an arm, CT, equal to CSi and CS 2 , 

 and always at right angles to OT : this arm bears the axis 

 through C. Then as S x and S 2 are guided along the corre- 

 sponding curves, the two arms OT and CT move in such a 

 manner that T is always the point in which a circle of radius 

 CSx, CS 2 , or CT is touched by OT; thus always OT 2 = OSi . 0S 2 . 



As regards the transmission of the motion of T to a point 

 S 3 on the axis OSiS 2 different methods are possible, perhaps 

 the simplest being that represented by the dotted lines in the 

 figure. ATBS 3 is a rhombus with hinges at the four angular 

 points ; the hinges at T and S 3 are attached to the collars 

 there, while the hinges at A and B bear two collars free to 

 move along an arm OBA, which in its turn is free to move 

 round the axis through 0, obviously always OS 3 = OT. A 

 pencil attached to the collar S 3 traces the required curve. 



All that is necessary, then, for finding the integral 



£ 



tt9 ^0)f(0)d0 



is to join the two ends of this curve by straight lines to and 

 take the area of the resulting figure by means of any planimeter. 

 But a very simple attachment to the above mechanism 

 makes a planimeter of it. Suppose the area of any figure to 

 be required. Fix the collar Si at a certain suitable distance b 

 from 0, so that when the arm OSiS 2 moves, Si must describe 

 a circle of known radius b. Gruide S 3 along the outline of the 

 figure, then S 2 moves always in such a way that 



JOS^ =i f OS! . O$ 2 d0=b$ OS 2 tf0 ; 



but this last integral is the length of the path travelled by S 2 . 

 Thus all that is necessary is to attach at S 2 a wheel which will 

 record the length of the path traversed by S 2 . 



The whole operation, then, of finding 1 $(Q)y$r(0)dQ 

 Phil Mag, S. 5. Vol. 20. No. 123. August 1885. N 



