for the Theoretical Proof of Avogadro's Law, 327 



In the latter functional determinant, besides the angles 

 already put constant, B' is to be regarded as constant. 

 Further 



a'=al+u\cosh, 



sin; : sin^=a : a'; 

 whence 



._ a sin h __ «sin/i 



n ^~ Vl — a" 2 — a 2 sin h ~" *\— oil cos h 



We see, further, from the figure that 



180°-K=A-<vW, 



when the latter angle depends simply on the form of the im- 

 pact, and is therefore to be regarded at present as constant. 

 So also 



; + 180=K'+<KVVi?. 



The latter angle again is constant ; whence it follows, since 

 nothing here depends upon the sign, that 



y ba; bk; , 9A *j 



Since in the equations for a! and tan j also the angle L, 



which equally depends only on the form of the impact, plays 



the part of a constant, the determinant can be calculated 



a' 

 without difficulty, and we obtain for it the value - . We 



might also have obtained this result without any calculation 

 by imagining the points v, v', V, and V as fixed. Since A 

 and h are spherical polar coordinates of the point X of the 

 spherical surface, so also A', j ; the element of area ad Adh 

 expressed by the former polar coordinates must be equal to 

 the element of area a'dA'dj expressed by the latter. We 

 have then 



a' dM dB' dK! = adA dB dK. 



For a fixed position of the points v, v', V, and V, A, K 

 and then A, K' may be regarded as spherical coordinates of 

 the point X, which would give at once 



xdA'dK=u'dA'dK'. 



Since, further, from the definition of A (equation 13), 



dv' dT dT dO' = Ldv dY dT dO, 



it follows from equations (28), (29), (30), (31) that 



d? dy' dlC &%{ d Vl ' d£ dO' __ ^'TVA 

 dgdfidtd&dyidkdQ ~ v 2 Y 2 t 



