1878.] On the Transformation of Elliptic Functions. 177 



Comparison of adjoining districts of one and the same side. 



Average percentage of times of Average difference of 



occurrence of superiority of temperature 

 temperature of one district over 

 another on both sides taken 



together 65-219 - 0-055° C. (0-099° F.) 



According to the above figures, superiority of temperature of a 

 given side of the head over the other, is a little less frequent of occur- 

 rence than superiority of temperature of a given tier or district of one 

 and the same side over another tier or district. The two sides of the 

 head are rarely equal in temperature, but the balance of superiority 

 shifts so frequently from one side to the other, that the percentage of 

 superiority for either side, in a given number of observations, is com- 

 paratively small. 



III. " Addition to Memoir on the Transformation of Elliptic 

 Functions." By A. Cayley, F.R.S., Sadlerian Professor of 

 Pure Mathematics in the University of Cambridge. Re- 

 ceived February 6, 1878. 



(Abstract.) 



I have recently succeeded in completing a theory considered in my 

 "Memoir on the Transformation of Elliptic Functions," Phil. Trans., 

 t. 164 (1874), pp. 397-456, that of the septic transformation, n=7. 

 We have here 



a solution of 



l — y_l—x/ a—(3x + ^x 2 —tx 3 V 



1 + IJ ~ 1 + x \a + fix 2 + -IX 2 + CXV 



Mdy dx 



1-vy Vl-x\ 1 



1 2(3 



where _=14-J_; and the ratios a : # : 7 : and the 'w-modular 



Ma 



equation are determined by the equations 



u\2 al + 2«/3 + (3-) =v\f + 2 7 3 + 2/33), 

 7 2 + 2/3 7 + 2ad + 2/3a=^t 2 (2a 7 + 2/3 7 + 2ac + (?) , 



C 2 +2 7 £ = vV°(a 2 + 2a/3) ; 



or what is the same thing, writing a=l, the first equation may be 

 replaced by S=— , and then, a, 8 having these values, the last three 



