beneath continents and oceans compared. Ill 



Multiply these equations by c n _!, c n _ 2 , ..., c 1} 1, where 

 c n -i> c,i_ 2 , ..., Cj, are arbitrary constants to be determined by 

 the condition that when the equations so multiplied are added 

 together as they stand, the n — 1 columns after the first shall 

 vanish, but not so the first. 



Then we shall have 



T, (V 1-1 + C l t 1 n ~* +...+ Cn_A + Cn-j) = Cn-J, 



t 2 (4 n_1 + cA n ~ 2 + ■■• + C n -A + c w _0 = 0, 



T 3 (*3 T" Ci&3 " + . . . + C w _ 2 fc3 + C n —i) = U, 



^"n V^w T CiCjj, " + ... + C n — 2 t n T n —i) — "• 



It appears therefore that the substitution of t 2 , t 3 , &c, £ n for ^ 

 causes 



Tl VI T Clfil + . . . + C n —2 ti + C n —i) 



to vanish. Hence this latter expression is by the theory of 

 equations equal to the product 



r 1 (t 1 -L)(t 1 -U)...(t 1 -t n ); 

 whilst it is also equal to c n _ T 6. Also 



C n — i = ( 1) t 2 t s • • • t n > 

 Hence 



TA («, - * 2 ) («l " «b) • • • (*1 ~ t n ) = (- l)^ 1 ^ . . . t n b. 



Similarly, by assuming another series of arbitrary factors 

 d 1} d 2 , ... d n - 1} such that the sum of every column except that 

 containing t 2 vanishes, we can prove that 



tA {k - k) (t 2 -t 3 )... (t 2 - t n ) = (- l)"-* tA ■ • • tnb, 



and so on. 



Consequently we have the n — 1 equations, 



tA (k - t 2 ) (tj. -t 3 )... (tj, - t n ) 

 = r 2 t 2 (t 2 - U) (t 2 -t 3 ) ... (t 2 - t n ) 



= T 3^3 \*3 *l) \*3 * 2 ) "• \*3 tn) 



&C. &C. 



= T n t n (t n tj) (t n t 2 ) ... \t n t n —i), 



each of these products being equal to (— l) n ~ 1 tA ... t n b. 

 Consider the equal products 



rA(t 1 -t)(t 1 -t,)(t 1 -Q...&c (1), 



= T. 2 t 2 (t*-t l )(t 2 -t s )(t 2 -t 4 ) ...&c (2), 



= r :i t :i (t s -t 1 )(t s -t. 2 )(t. i -t i ) ... &c (3), 



= T 4 * 4 (*4-* 1 )(*4-* 2 )(*4-*8)...&C (4). 



