NATURE 



[April 15. 1920 



different degrees, and it becomes a definite problem 

 to determine for any muscle or combination of muscles 

 the relation between the speed of muscular contraction 

 and the muscular force which will yield the greatest 

 external power. The problem may be solved by means 

 of the diagram in Fig. i. 



It is assumed that the muscular machine has a con- 

 stant output of power which is represented by the 

 product xy (x pressure, y velocity) of the co-ordinates 

 of the hyperbola AB. Also, that the pressure avail- 

 able for producing exterior power is less than x bv 

 some quantity V'fy) depending on the velocity, and 

 represented in the figure by the abscissa of the curve 

 DD', the effective velocity in the same way being less 



Fig. 1.— X,Y, co-ordinates of hyperbola AB. <;,(.r), ordinales of CC. 

 ^{y), abscissa of DD'. 



than y by some quantity <^{x) depending on the pres-' 

 sure, and represented by the ordinate of the curve CC. 



The useful power is evidently (.r ->//•(>/) )(v - 0(;i')), 

 and the loss of power is .r0(;t-)-|-^^(v)-0(;r)\/^(y). 



If the co-ordinates for x->rdx and y + dy are drawn 

 as in the figure, it is plain that the loss is a minimum 

 (and the useful power, therefore, a maximum) when 

 {x-y^(v) )d<i>{x) + (y - <l>{x) )dy^{y) = ,/,(_y) dy + (j>{x)dx. 



I know of no experiments which would determine 

 the form of the functions <p and V— that is, what 

 power is lost in sustaining a load or in keeping a 

 uniform speed. Both these subjects are worthy of 

 mvestigation, and, with the facilities offered bv s'ome 



Velocity 



y 



Kesistancc 

 (—(Pi 



Velocity 



External Force. 



leaJrage 



<PW 



of the modern laboratories, ought not to present anv 

 great difficulties. 



If '^{x) and "^(y) were simply proportional to x 

 and y, the most economical speed would rjiake, if 

 4>(x)=ax and ^iy) = dy.y/x = d/aj and if a = d, the most 

 economical speed would be the mean between that where 

 ylriy)=x and (p{x)=y. 



In reality, however, \^(y) is, I believe, much less 

 than <t>{x), but this remains for experimental deter- 

 mination. It may be noticed that as ^(v)/4>{x) de- 

 creases, the most economical speed increases. 

 NO. 2633, VOL. 105] 



A close analogy to the conditions of the problem 

 may be found in a fluid contained between two pistons 

 A and B (Fig. 2), between which there is a leak 

 governed by the fluid pressure. A constant power 

 urges A towards B, and A itself is subject to a fric- 

 tional resistance depending on its speed. The useful 

 work is represented by the velocity of B against r,n 

 exterior force, while the leak stands for ^(x) and the 

 frictional resistance of A for yp{y). 



A. Mallock. 



A Dynamical Specification of the Motion of Mercury. 



If we assume that the modified Lagrangean func- 

 tion for two mobile and massive particles is of the 

 form 



L = |;«i(.r,2 -t-ji), 2 -f ^1^) + i,,n,(x./ + y^ + z,?-) 



■ r L (?^ 



(.r,-x,)2 + (ri-j/.,)2 + (^ 



,-..)] 



where the symbols have the usual meaning, C being 

 the velocity of light and A a pure number, then the 

 principle of least action hfLdt^o leadv- to the following 

 conclusions : 

 (i) The motion of the centre of mass is constant. 



(2) The orbits of the two particles about their centre 

 of mass are similar and similarly described plane 

 curves, and independent of the motion of the centre 

 of mass. 



(3) For the orbital motion the modified Lagrangean 

 function is : 



Hence the equations : 



y{mi + m^) 



and 



r 2(m, + m^)\y 



1 + 



ra 



}r^0-- 



Writing r = i/u, we have: 

 /du^ 



'■ I ^2^2y{ m,+m^ )f 



h^ \ 2a) ('+ C^ I 



From this we may deduce 



■jn^\ yiPh + w^J 'KyK/ny + >Ji.,j \ 

 ' ' C'a ) 



de^ V"<z^ h^ )- h^ (' 



or the solution in the form 



11 = - jl+^COS (/it^-l-'?)]- 



where 



4 Xyg(;«i+w., )2 



These equations are exact. 



In applying this argument to the observed apsidal 

 progress of the planet Mercury, it is to be noted that 

 the interpretation of a and h differs slightlv from what 

 it would be if A were zero ; but to a sufficient degree of 



