CONSTITUTION OF MATTEE AND ANALYTICAL THEORIES OF HE AT. 41 



Now, let f^{x) and V^{x,t) stand for the auxiliary functions Y{x,0) and 

 Y{x,t) of (i). Then, with the same initial temperatures as in (i) , amj function 

 of the form 



4- 1 \ ^ a" COS (c"a;) 



1 ^ 



may be taken as the initial auxiliary function, d being any constant whatever. 

 And the Solution of the problem is furnished by Fj {x, t), whatever d may be, the 

 degree of the approximation, however, depending on f7. Thus, when 0 <\d\<l, 

 p{SJ E+e) ^^j^ ^ j^^^ ihsin 6 x 10~^ or 2 X 10""^ according as A is 



taken equal to x 10"' or 3"^ x 10"^ respectively. 



2 2 



(iv) Suppose that = IQ-'X^, —<K < T' A 3"' x lO"' x k\. 



Let Y{x,0) — f (x) = x'^", n>l. Then the left side of (Pj) is less than 



27tf [ 



|43/' (.) + 1 f" (.) (67 + < ''-(^- + ')'^ 1 145 + 136. 



<3>7(W + 1) (27^ + 1)2,. 



Therefore -^^(^^+^) well as — is less than (" + '^^ + ^ ) '^i ag 

 // (j^ 1 — 5^ (w + l) (2» + l) Aj 



long as > 7?. (n + 1) (2^^ + 1). 



The foUowing ^at^e shows clearly how rapidly the error may increase as 

 n increases , i.e., as the change of initial temperature from one assemblage to 

 another increases: 



n 



Superior 



limit 



Superior limit of error. 



Ol error. 



A = 3-^ X 10-' 



A = 3-^ X 10-« 



A = 3-^ X 10-" 



1 



18AJ1- 



6A, 



6 X 10-^ 



2 X 10-' 



2 X 10-^ 



1.25 



30AJ1- 



m. 



101x10-^ 



^ X 10-' 



3J X 10-^ 



1.5 



45A,/1 - 



15Aj 



15i X 10-^ 



5 X 10-' 



5 X 10-^ 



1.75 



66A,/1 - 



22A, 



23 X 10-' 



7 X 10-' 



7 X 10-^ 



2 



90A,/1 - 



30A, 



32i X 10-' 



9A X 10-' 



9i X 10-^ 



3 



252A,/1- 



84A, 



greater than 1 



28 X 10-' 



3 X 10-' 



4 



540AJ1 - 



180A, 





66 X 10-' 



6 X 10-' 



5 



990A,/1- 



330A, 





greater than 1 



10 | X 10-' 



11 



9108A,/1- 



3036Aj 







greater than 1 



händig. 



d. K. Ges. d. Wiss 



zu Göttingen 



Math.-Phys. Kl. N. F. 



Band. 2,4. 



6 



