Ball — On Somographic Screiv Systems. 441 



this special type is also of importance for other kinetic problems, it 

 will be desirable to examine into its general character. 



If the general linear transformation, which changes each screw a 

 into its correspondent 6, be specialized by the restriction that the 

 co-ordinates of 6 are given by the equations 



\dU 



Pi (iai 



Pz was 

 &C., 

 VdV 



PedoLs 



where Z7 is any homogeneous function of the second order in aj, . . . a^, 

 and where pi, . . . pe are the pitches of the screws of reference, then 

 the two systems are related by the special type of homography to 

 which I have referred. 



The fundamental property of the two special homographic systems 

 is thus stated : — 



Zet a and (S he any two screws, and let 6 and <^ he their correspon- 

 dents, then, when a is reciprocal to cj}, (3 will be reciprocal to 6. 



"We may, without loss of generality, assume that the screws of re- 

 ference are co-reciprocal, and in this case the condition that (i and & 

 shall be co-reciprocal is 



PiMi +P2M2 - • ■ • +i?6A^6 = ; 



but by substituting for 6^ . . . 0^, this condition reduces to 



aa. da 



Similarly, the condition that a and <^ shall be reciprocal is 



dU dU 



dpi d/3e 



It is obvious that as His an homogeneous function of the second de- 

 gree, these two conditions are identical, and the required property has 

 been proved. 



It is easily shown that by suitable choice of the screws of reference 

 the function C^may, in various ways, be reduced to the sum of six 

 square terms. We now proceed to show that this reduction is pos- 



