520 KEPOKT— 1894. 



exhibited a somewhat improved form early this year at the Institution of 

 Civil Engineers. 



He has given the rod between the knife-edge and the tracer the 

 shape of a circular arc, radius I, and engraved a scale on it, so that it is 

 possible to measure the arc between the two marks made on the paper 

 instead of the chord. 



Quite recently (October) I have learnt from Coradi that in December 

 1893 F. Hohmann communicated the idea of such a planimeter to him, 

 and also that Professor Ljubomir Kleritj, of Belgrad, had invented a 

 new planimeter. This is only a modification of Prytz's. In it the 

 knife-edge is replaced by a knife-edge wheel, whilst the other end rests 

 on two feet between which the tracer is so placed that its point is just 

 off the paper. These feet are fastened to a cross-bar which is movable 

 about a point above the tracer. The instrument thus rests firmly on 

 three points. Both Professor Unwiri and Coradi have pointed out to 

 me that it would be an improvement if to Prytz's form a disc be added 

 as a handle which can turn freely about the tracer. Kleritj 's form 

 supplies this. 



Just now I have received a reprint from a Servian journal in which 

 Professor Szily, of Budapest, has, at the request of Kleritj, developed the 

 equation to the path of the knife-edge when the tracer moves along the 

 circumference of a circle. It is dated December 1893. But it does not 

 seem to advance the curiously interesting theory of the instrument. 



The geometrical considerations given above were started by me and 

 more fully worked out by Mr. A. Sharp, who has obtained several other 

 i-esults, which, however, do not yet yield any better practical rules than 

 those given by Prytz. 



Linkage Integrators. 



J. Amsler (' Vierteljahrsschrift,' 1856, p. 29) describes what he calls a 

 ' Flachenreductor ' (area-reducer), and in connection with this he gives a 

 theorem about pantographs which deserves notice. 



Starting with the fundamental theorem about the generation of an 

 area by a line of finite length, these theorems are easily obtained. 



If we denote by (A B) the area swept over by the line AB during a 

 cyclical motion, by (P) the areas enclosed by the close path of a point P, 

 then we have 



(AB)=(B)-(A) + «7rAB2 



If /i=0, and this case alone we shall follow up, we have 



(AB)==(B)-(A) 



If we take a third point C on, or off, the line AB, we always have 



(AB)=(B)-(A), (BC)=(C)-(B), (CA)=(A)-(C) 



These give at once 



(AB) + (BC)-1-(CA)=0 and (AB) = (AC)-f(CB) 

 where account always must be taken of the sense, viz., it is 

 AB = -BA and (AB)= -(BA) 



