200 



Capt. M. Greenwood. 



is, by Glazebrook and Dye's formula, a linear function of the external 

 work performed. For a constant performance of work, the change of heat 

 production with changing mass is described by a hyperbola, one asymptote 

 of which is parallel to the axis of H (for W = 0, the equation is that of the 

 asymptotes), and, in consequence of the position of the principal axes with 

 reference to the axes of H and M, the minimvim of H moves to the right 

 with increasing values of W. 

 In effect, we have 



dU (« + /3M)2' ^ ^ 



(4) 



dM^ («4-/3M)3' 



(4) is always positive and from (3) the value of M for a minimum is 



M = -ci/^±^(W/bl3). 



This decreasing disadvantage, or, rather, increasing advantage, of mass 

 with increase of work is in agreement with the remarks of Prof. Macdonald,* 

 and, pro tanto, is an argument in favour of the form of Glazebrook and Dye's 

 expression. But that the formula itself is no more than an interpolation 

 formula is manifest, since it gives negative values of H for small values of M 

 when W = 0, while, even within practicable ranges, it leads to the 

 paradoxical result that the absolute heat production associated with the 

 performance of 56 thermal units of work in a mass of 20-30 kgrm. is not 

 less than that of a mass between 50 and 60 kgrm. I thought, therefore, 

 that it would be of interest to determine whether a linear interpolation 

 formula might not reproduce the experimental results with an accuracy 

 comparable with that achieved by G-lazebrook and Dye's second degree 

 expression. Taking the data on p. 313 of Glazebrook and Dye's paper, I 

 computed the various constants of a first degree multiple regression 

 equation ; these are set out in Table I, and the deduced eqiiation is 



H = 24-900 + 3-940 W + 1-755 M. (5) 



Testing this against the observations, I* obtain Table II, which is a 

 reproduction of Glazebrook and Dye's Table III, with the addition of the 

 values computed from the above regression equation. It is easy to see that 

 the linear expression is not inferior to Glazebrook and Dye's expression in 

 its power to reproduce the experimental results, although it has one less 

 constant (actually, the mean square of error is rather smaller in the present 



* Op. cit., p. 111. 



