SIR Gr. B. AIRY ON THE TIDES AT MALTA. 
129 
Substituting the numerical values for the trigonometrical symbols, 
Gl=M+ ij63 X ( +1_ °) + rl63 X ( 0+1 > +: VM8 X < +1- ' / ^ +:i m8 X ( 0+v/i) ’ 
G2=M+ I^3 x(0+1 ) + TjM x ( -1+0)+ 7848 x GK / i-°)+.;^x(-v / i+ 1 )> 
G3=M+j^L3X(-l-0)+j^X(0-l) + ^x(0+ v / i)+^x(-l + v /4), 
G4=M+ 1 -^x(0-l)+ i ^x(+l+0)+^x(- v /i+l)+:^x(-Vi+0), 
G5=M+ r^63 X ( +1_0) + lj63 X ( 0+1)+ -' i 7M8 X ( -1 + ' / i) + : 7M8 X ( 0- ' / W’ 
G6=M+ rj63 X ( 0+1 ) + I : M63 X < _1 + 0 ) +; 78i8 X < _ ' / ^ +0 ) +; W8 >< ( + ^ _1 )’ 
G 7 = M + 1^63 X ( _1 _0) + m 63 X ( 0 _ 1 )+; 7 M 8 X (0 ~ ^+ 7^8 X <+ 1 “ 
G8=M+ rle3 x ( 0-1 ) + ri63 x < +1+0 )+7Mi x ( + ^ -1 ) +; 7M8 x ( + ^ -0) - 
As we have here eight equations from which five quantities are to be determined, it 
is proper to refer to the considerations of the Theory of Probable Errors. Put e for 
the numerical value of each of the probable errors e 1} e. 2 , . . . <? 8 of the numbers Gl, 
G2, . . . G8. Now to form the final equations, determining p and q, we ought in 
strictness to form intermediate equations by multiplying the equations above by the 
coefficients of p and q in those equations ; and thus we should obtain results for 
. and whose probable errors are 
But if, instead of this, we had 
•7848 -7848’ r ^/(S- W ¥ )' 
multiplied each equation by +1 or by — 1 so as to make all the coefficients for p 
positive (and similarly for a), we should have obtained results for and 
‘7848 "784o 
whose probable errors are This is greater than that found by the first process 
by only part. The loss of accuracy in this second process is so small, and the 
gain of convenience and simplicity so great, that I have not hesitated to adopt it. 
W e now obtain the values of M, P, Q, p, q, by the following combinations : — 
Gl+G2+G3+G4-fG5 + G6fi-G7+G8 — 8 M, 
Gl + G2-G3-G4+G5 + G6-G7-G8= I |^, 
Gl — G2 — G3 + G4+G5 — G6 — G7 — G8 = ^5^63* 
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