TRANSACTIONS OB THE SECTIONS. 19 



the increased reading of the hand on the dial is thus = Xdx-\-Ydi/ ; and supposing 

 the original reading to be zero, and the driver D to he moved from its original post" 

 tion along an arbitrary path to any other position whatever, the reading on the 



dial will be the corresponding value of the integral [(Xd.i + Ydy). 



It is obvious that we might by means of a combination of two such mechanisms 

 calculate the value of an integral J/(«) du along an arbitrary path of the complex 

 variable u,=x+iy; in fact, writing /(*+(v) = r+e'Q., the differential is now 

 (P+iQ)(dx + id!/), = rdx — Qd!/+i(Qdx'+Vdi/); and we thus require the calculation 

 of the two iutegrals |"(P<fa— Qdy) and UQdx+Tdy), each of which is an integral 



of the above form. Taking for the path a closed curve, it would be very curious to 

 see the machne giving a value zero, or a value different from zero, according as the 

 path included or did not include within it a critical point ; it seems to me that this 

 discontinuity would really exhibit itself without the necessity of any chaDge in the 

 setting of the machine. 



The ordinary modes of establishing a continuously variable velocity-ratio between 

 two parts of a machine depend upon friction ; and in particular this is the case in 

 Prof. James Thomson's mechanical integrator : there is thus, of course, a limitation 

 of the driving-power. It seems to me that a variable velocity-ratio, the variation 

 of which is practically, although not strictly, continuous, might be established, by 

 means of toothed wheels (and so with unlimited driving-power), in the following 

 manner : — 



Consider a revolving wheel A, which, by means of a link BC pivoted to a point B 

 of the wheel A and a point C of a toothed wheel or arc D, communicates a recipro- 

 cating motion to D, the extent of this reciprocating motion depending on the dis- 

 tance of B from the centre of A, which distance, or say the half-throw, is assumed 

 to be variable. Here, during a half-revolution of A, l) moves in one direction, say 

 upwards ; and during the other half-revolution of A, D moves in the other direction 

 say downwards, the extent of these equal and opposite motions varying with the 

 throw. Suppose, then, that D works a pinion E, the centre of which is not abso- 

 lutely fixed, but is so connected with A that during the first half-revolution of A 

 (or while D is moving upwards) E is in gear with I), and during the second half- 

 revolution of A (or while D is moving downwards) E is out of gear with D • the 

 continuous rotation of A will communicate an intermittent rotation to E, in such 

 manner, nevertheless, that to each entire revolution of A, or rotation throuo-li the 

 angle 2tt, there will (the throw remaining constant) correspond a rotation of E 

 through the angle n . 2ir, where the coefficient n depends upon the throw. And evi- 

 dently if A be driven by a wheel A', the angular velocity of which is r- times that 



A 



of A, then to a rotation of A' through each angle — there will correspond an entire 



revolution of A, and therefore, as before, a rotation of E through the determinate 

 angle n.2w; hence, X being sufficiently large, to each increment of rotation of A' 

 there corresponds in E an increment of rotation which is tiX times the first-men- 

 tioned increment, viz. E moves (intermittently, or in beats as explained, and possi- 

 bly also with some " loss of time " on E coming successively in gear and out of 

 gear with D) with an angular velocity which is =MA times the angular velocity of 

 A-' : and thus the throw, and therefore n, being variable, the velocity-ratio nX is also 

 variable. 



AVe may imagine the wheel A as carrying upon it a piece L sliding between guides, 

 which piece L carries the pivot B, of the link BC, and works by a rack on a toothed 

 wheel a, concentric with A, but capable of rotating independently thereof. Then 

 if a. rotates along witli A, as if forming one piece therewith, it Avill act as a clamp 

 upon L, keeping the distance of B from the centre of A (that is, the half-throw) con- 

 stant ; whereas if a. has given to it an angular velocity different from that of A, the 

 effect will be to vary the distance in question — that is, to vary the half-throw, and 

 consequently the velocity-ratio of A and E. And in some such manner, substitutino- 



