﻿the Rigidity of the Earth. 223 



Now, by a known theorem, if 



D = F = 0, 

 the moments of inertia about the new principal axes are the 

 roots of the equation 



(H-A)^ + (H-B)i/ 2 +(H-C)^ + 2E^=0, (V.) 



where A, B, C are the given moments about the old axes. 

 In the present case we may put 



A=B. 



Now, if the angle between the axis of the new greatest 

 moment of inertia and the axis of z is 0, then by the trans- 

 formation 



x = x l cos6 + z 1 sin#, 



jgrss. — Xi sin 6 + z ± cos 0, 



we may easily find the angle from the formula V. We 

 obtain 



tan 20=-^ (VI.) 



But by definition the angle ( < POP) is equal to f- of 

 the angle POR, and 



c = cos (POR) 



a = sin (POR) 



As the angles POR and POP' are very small, their cosines 

 are nearly equal to unity, and their sines are nearly equal to 

 the arcs ; but the arc POR is equal to \ of the arc POP' 

 [the arc 0]. Hence, neglecting small quantities of second 

 order, we obtain from the formula VI. 



* 2 'P u * , . . . (VII) 



7 "ldn + VgpR' 0-0011 K } 



But Prof. Newcomb thinks that one fourth of the angle 

 POP' may be attributed to the influence of the Ocean. 

 Further, as the product of inertia E standing in the nume- 

 rator of the right-hand member of VI. depends principally 

 on the deformation of superficial layers, and for that reason 

 the mean density in E must be smaller than the mean density 

 of the Earth — we must multiply the right-hand member of 

 Vll. by a factor smaller than unity. We take, with Prof. 

 Newcomb, the mean effective density to be 0*6 of that of 

 steel, i. e. 4*68, and multiply the right-hand member of VII. 



with _ . It is to be remarked that by the meaning of 



0*0 



