Mr Harvey on the Method of Minimum Squares. ^95 
confine the extreme errors within the least possible limits, with-» 
out attending to the nature of their signs. 
Of all the principles that can be proposed for the attainment 
of this object, there is none more general, more exact, or of an 
easier application, than that which consists in making the sum of 
the squares of the errors a minimum. By this method, there is 
established among the errors a kind of equilibrium ; and which, 
by preventing the extremes from prevailing, renders it very pro- 
per for making known the state of the system which approxi- 
mates the nearest to the truth. 
The sum of the squares of the errors E^, E'^, E"^, &c. being 
(« -h 5 a? -f -f -f &c.)^ 
-j- {a! + h'x 4- c^y -{- -p &c.)^ 
4- {d' 4- Vx 4- d'y 4- f'^z 4- &c.)^ 
4- &c. ; 
if we endeavour to obtain its minimum^ by making x alone to 
vary, we shall have the equation 
0 = ab X J b^ J he z ( h /4- &c., 
in which the sum of the similar products a 5, a' 5', a" 5", &c. is 
denoted by fa b, the sum of the squares of the co-efficients of 
X ; that is to say, + b'^ + b"^ -f &e. by fb\ and SO on. 
The minimum with relation to y^ will give similarly, _ 
0 = J ac + xj bc + ^ f (^ + z jf 'e + kc., 
and the minimum with relation to 
0 = j afyx j bf-^y J Cf+Z Jf^ + kc., 
where we perceive that the same co^efficients fb c, fbf, &c. are 
common to two equations, — a circumstance which facilitates the 
operations of the calculus. 
In general, ^ if be required to form the equation of minimum 
with respect to one of the unknown quantities, it will be necessary 
to multiply all the terms of the proposed equation by the co-efh 
cient of the imiknown quantity in this equation, tahen with its pro^ 
per sign, and then to find the sum of all these products. 
In this manner, we shall obtain as many equations of mini^ 
mum., as there are unknown quantities, and these equations it 
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