558 REPORT — 1894. 



the conditions which must be satis6ed in order that a binary quantic of this 

 degree 2n may be a perfect square, and show that they may be all found from a 

 matrix which I call the square matrix for the functions of the degree 2n. ^ I have 

 not entered on any discussion of these curious conditions and their intimate 

 relationship, which are well worthy of examination, insomuch as their number is 

 the number of ways in which 4n-3 quantities may be taken 2n — 2 together, 

 and are still equivalent to but n conditions. 



The Abelian system of differential equations may be written 



_-^ s'VZs 



where there are 7» quantities Sj, s^, s^, and wi-1 equations, as is clear from the 

 aboTe method of writing them if we suppose that i can have any integer value 

 from I = to i= m - 2 ; also /(s) = s"" + Pis""-' + P,s^™-2 + . . . P,„,. 



I now form a function which I call F {£), and which is of the degree 2»i - 2 

 in the following manner. 



Let 



<^(s) = (s-2.)(s-s,) . . . (s-s„,) 





*m T 



L (=)=='" + Pi s™-i + X,s'"-- + X3S'"-*+ . . . X„ 

 L (c) introducing m — \ quantities X^, Xj . . . X„i ; 

 then I vcrite 



F(=) = Ao {/(=) + {(/. (=)} ^ - •2<i> {£) L (.) } , 

 which is easily seen to be of the degree 2m -2 in s; also its source is 



K \pm - 2XmPm + Pom | • 



Now I say this function F {z) must be a perfect square. Forming, then, the various 

 conditions from the square matrix of F (:). we have all the forms of the algebraic 

 inteorrals of the Abelian system 



^ =0, 



^s/Ao/(=) ' 



which are rational and integral, involving m — 1 arbitrary constants \, X^ . . . X„,. 



2. On a Graphical Transformer} By A. P. Trotter. 



This instrument is intended for the expeditious replotting of a curve with 

 transformed ordinates without calculation or scaling. It consists of a rectangular 

 frame and a curved template or cam, and is used in conjunction with a straight 

 ruler. 



Let the scale of one system of ordinates be set off^ upwards along the edge of 

 one of the perpendiculars, and the scale of the other along the edge of the other 

 perpendicular, but downwards. Join the corresponding points on the scale by 

 straight lines. The envelope of this system of lines may be thus drn-wn, and to this 

 curve a cam is cut in thin wood or ebonite. 



To transform any ordinate, set the frame against a T square, adjusting the edge 

 to the ordinate, and the zero to the zero of the scale. Set a needle at the extremity 

 of the ordinate ; bring a straight edge to touch the needle and the cam ; prick off' 

 a point at the intersection of the straight edge with the other edge of the frame. 

 This point determines the length of the new ordinate. 



An instrument provided with a logarithmic cam was exhibited. With this 

 instrument the product or quotient of two curves can be found by adding or sub- 

 tracting the logarithms of the ordinates ; or the logarithms of a series of observa- 

 tions can be plotted. Cams for other functions can be easily made ; but it must be 



' Printed in extenso in the Electrician, August 17, 1894, vol. sxxiii. p. 465. 



