538 
MR. HENNESSY’S RESEARCHES IN TERRESTRIAL PHYSICS. 
loping the irrational factor in a converging series, and then integrating separately 
the resulting terms. For greater uniformity I develope in all the integrals the irra- 
tional factor, which becomes when developed 
l-}-|g^sin^ ^-)-|*^£^sin*^-}-|-^-|g®sin®^+ etc. 
Hence the sum of the four integrals will be, neglecting terms of the order 
^i[cos ^2 / cos 6 sin — sin 0^/ cos^ 0dd^ cos 6.2/ cos^ 0 sin sin 0^/ cos* 0d0 
— 2®^{^i[cos 02/ sin® 0 cos 0d0 — sin 02/ cos^ 0 sinW^j -1- cos^^/* cos®^ sin® 0d0—sm02/ co&*0 sin 
On taking these integrals between the limits 4 0 ^, the expression becomes 
Aj j^^^^(cos 2^2 — cos 2flg) — 2^3— sin 2S^) + fig — fig J '^^2 ~ h + cos 2^2— cos 2fi3j 
_s^[l^^.^^g^_^.^^g^^_^^.^2g^_sin2fi2 + |(fi2-fi3)]-^£2|^i^^[^(cos4fig-cos4fl2) 
- (cos 2fig - cos 2fi2) J I" 1 45^ _ g-^ 4y _ J c^ |- 1 
- 3 (cos 2flg - cos 2fi2)^ + (sin efig - sin 6fi2) + i(sin 45g - sin 4fi2) -^(sin 2fig - sin 2fi2) - 2 (,fig - fig) J J 
= [(fli) sin (fi 3 +fi 2 )sin (fis— fig) + (« 2 ) 2 (fi 3 + fi 2 ) sin 2 (fi 3 — fi 2 ) + (flg) sin 3 (fig + fig) sin 3 (fig— fi 2 )] cos fig 
- [(61) cos (fig + fig) sin (fi 3 — fi2) + (^2)003 2 (fig + fig) Sffl 2 (fig — fig) + (^g) COS 3 (fig + fig) Sm 3 (fig — fig)] Sm fig > . 
-(Ci)(«3-y sinfig. 
Making for brevity 
[*'+2“8't*‘ + 8) 
1 J J ^3. — 16(^^ 2^ ^^0’ ^® — 
(<^ 2 ) = Ye [ ^ + 2 )] ’ 
(Ci) — 2 [^^1+4 + ^"2)] ■ 
( 38 .) 
20. I now proceed to determine the expressions fory^/®c?L. In general the cohesive 
strength of an unit of surface of the section of rupture may be considered proportional 
to the number of molecules of which it is the section, and as the density fg must at any 
point be also proportional to the number of molecules at that point, it follows that 
A- = 5,7% 
(§2) 
The section of rupture may evidently be considered as a trapezoid, hence 
<^L=— (r2+/) cos ^2<//=— (1— 2® cos®^) cos + cos02dl 
T j €L-^ ^ j 
very nearly. Hence 
