AET. 4.] SUPPLEMENT TO CHAP. XHI. 247 



while for M 8 we have at once, 



M =- A Cf + B Cr +l _ _ A % + B fr^ 



Cs-l Zs-1 + 2 CB (Z 8 _! + Z 8 ) C g+1 Z B 



If, in like manner, we should multiply the last of equations (6) by the 

 number di, the last but one by d 3 , the rth by ^ a -r+i> etc. ; then add, and 

 make all terms, except that containing M a , equal to zero ; we should have 

 the conditions: 



2 dt (k + Zs-i) + da ZB_I = ; 



d Zg_i + 2 dt (Zg_! + Zg-s) + d t Zg_ 8 = ; 



+ 2 <Z 8 _ r +2 (?r + ?r-l) + ^s-r+S ?r-l = 0; 



^s_ a Zs + 2 dg.! (Z 8 + k) + <?B Z a = ; 

 while for the moment we have 



M 3 = At? a-r+2 + B (Z 8 -r+l _ _ A<Zs-r+8 + B <?a-r+l ^ 

 ^s_l Z a + 2 d a (It + Zi) (Z 8 +l Zi 



The values of M 2 and M 8 are thus given in terms of the quantities 

 A and B and c and d. 



A and B depend simply upon the load and its position in the rth span. 

 Thus A = P l t \2 Tc - 3 F + F), B = P l t \le - * 8 ). 



As for the multipliers c and d, they depend only upon the lengths of the 

 spans, and need only satisfy the conditions above. Hence, assuming 

 d = 0, ct = 1, and d l = 0, <Z a = 1, we can deduce the proper values for 

 all the others. Thus, 



^ = Oj di = 0, 



fa = 1, d, = 1, 



_ ?1 + ? ,7 _ O ? B + k-1 



* S= - - S A - , 



Q- - / Q/ - - . - 



C 4 = A Ct - -- a y, 4 = -o i - - j -- a , , 



It t> <a_2 *g 2 



/. 

 Ct - 



0/7 -3 

 =r xJ a& 



etc., etc. etc., etc. 



Now from equations (6) we see at once that M 8 = c* M 2 , M 4 = c 4 M a , 

 etc., or, universally, when n < r + 1, 



M n = Cn M 9 =- *- (Ad^^ + B^,^) . . (7) 



Also taking the same equations in reverse order, M,^ = <Z t M 8 , Mg_3 =: 

 2i M B , etc., or, universally, when n > r, 



M. = -J. (A^ + Bev+l) ... (8) 



