70 Mb. HOPKINS, ON THE MOTION OF GLACIERS. 



of which the temperature is V + C cos UttI + '^\ when this quantity is negative, and zero for 



those values of t which render the expression positive. If K = the first of these conditions will 



1 3 1 



be satisfied from ^ = to ^ = - , from t = I to t = - , &c. ; and the second from t = - to i* = 1 , 



2 -^ 



from t = - to < = 2 &c. If y do not = 0, the former of these periods will be shortened and 



o 



the latter lengthened, or the converse, according as V is positive or negative ; if, however, V be 

 small compared with C, the periods will be approximately as above stated, and such, therefore, 

 we shall consider them. They will be semi-annual, if we take one year as the unit of time. 



The theorems given by Poisson, in his Thiorie de la Chaleur, Articles 194, 195 ^and 19(), will 

 enable us to obtain the required solution. 



23. If the external temperature be represented by the general formula 



B + A coi (mt + e) + A^ cos (Wj^ + e^ + As (cos m^t + e^) + &e (l), 



and u denote that part of the internal temperature which depends on the external, we shall have, 

 at the depth .r beneath the surface, 



b . --\/^ ( a: fm A 



B +rir Ae " -cos(m;f+e V 6 



^ \ a '2 I 



D 



+ i.^,r^V^cosU. + e.-^\/^-^.) 



+ &c (2). 



Where D cos S = 6 + -V—, Dsmh = - 



a 2 an 



„, ,, b\/2m m 



and therefore D' = b" + — + -; 



a a' 



k 

 with similar formulae connecting £>,mi^,, D^nii^^t &c. Also a' = - , where k represents the 



c 



conductivity and c the specific heat of the matter constituting the globe ; and 6 is a quantity 



depending on the conductivity and radiating power of the surface. 



Now generally if (p (t) denote any function whatever, continuous or discontinuous, whose values 



recur whenever t is increased by 6, so that (p (t + 0) = (p (t), we have the general formulae 



<p(.t)^j:<i,{t')dt' 



^ r^ ^ ,.i^ 27r<' . 27r< 2 /«,,,> . 2t<' , , . Zirt 



+ el '^^'^'°'~0"'^'-'°'T~ + 0-/o ^(OBin-^d^.sin — 



+ &c. 



which will coincide with (l) when the following equations are satisfied, 



27r 47r Btp 



ra = — , m, = — , m^ = — , &c. 



B=\^l<p(f)de 



