18 REPORT — 1877. 



When the orbit is parabolic, a becomes infinite ; and since r-\-r' and k are finite, 

 the quantities a. and (i become indefinitely small. 

 Hence 



r+ r>+k =1 _ cos ^ +a ) = i(js +a y ultimately, 

 2a 



r+r'-k =1 _ cQS ( /3 _ a )_i( / 3_ a )a ultimately; 



2a 



also 



t= (-)*{/3+a- sin (/3+a)-(/3-a)+ sin (/3-«)} 

 = (-)V[(^+«) 3 -K/2-a) 3 } ultimately 



4r)Mf^-(^)*} "^ 



b V /a 

 which is Lambert's theorem in the case of the parabola. 



Suggestion of a Mechanical Integrator for tlic calculation of \(Xdx+Ydy) 

 along an arbitrary Path. By Professor Caylet. 



I consider an integral \(Xdx-\-Ydy), where X, Y are each of them a given func- 

 tion of the variables (x, y). Xdx-\-Ydy is thus not in general an exact differential ; 

 but assuming a relation between (x, y), that is, a path of the integral, there is in 

 effect one variable only, and the integral becomes calculable. I wish to show how 

 for any given values of the functions X, Y, but for an arbitrary path, it is possible 

 to construct a mechanism for the calculation of the integral, viz. a mechanism such 

 that a point D thereof being moved in a plane along a path chosen at pleasure, the 

 corresponding value of the integral shall be exhibited on a dial. 



The mechanism (for convenience I speak of it as actually existing) consists of 

 a square block, or inverted box, the upper horizontal face whereof is taken as the 

 plane of xy, the equations of its edges being y=0, y= 1, x=0, x=l respectively. In 

 the wall-faces represented by these equations we have the endless bands A, A',B,B' 

 respectively, and in the plane of xy a driving point D, the coordinates of which 

 are (x, y), and a regulating point R, mechanically connected with 1) in suchwise 

 that the coordinates of R are always the given functions X, Y of the coordinates of 

 D* ; the nature of the mechanical connexion will, of course, depend upon the par- 

 ticular functions X, Y. 



This being so, D drives the bands A and B in such manner that to the given 

 motions dx, dy of D corresponds a motion dx of the band A, and a motion dy of the 

 band B ; A drives A' with a velocity-ratio depending on the position of the regu- 

 lator R in suchwise that the coordinates of R being X, Y, then to the motion dxot 

 A corresponds a motion Xdx of A' ; and similarly B drives B' with a velocity-ratio 

 depending on the position of R, in suchwise that to the motion dy of B corresponds 

 a motion Ydy of B'. Hence to the motions dx, dy of the driver D there correspond 

 the motions Xdx and Ydy of the bands A' and B' respectively. The band A' drives 

 a hand or index, and the band B' drives, in the contrary sense, a graduated dial, 

 the hand and dial rotating independently of each other about a common centre ; 



* It might be convenient to have as the coordinates of E, not X, Y, but %, q, determi- 

 nate functions of X, Y respective!}'. 



