S96 Mr Harvey on the Method tf Minimum Squares, 
will be necessary to resolve by the ordinary methods. But we 
shall find it necessary to abridge the whole of the computation, 
by admitting into each operation, only as many figures, either 
whole or fractional, as the degree of approximation of which the 
question is susceptible, may require. 
If it were possible to satisfy all the equations of condition, by 
rendering each of the errors null^ we should equally obtain this 
result by the equations of minimum : for, if, after having found 
the values of a*, &c. which render the errors E E', &c. ze- 
ro, we cause £c, y^ z, &c. to vary respectively by ^ a;, ^y, ^ z, &c., 
it is evident that E^, which was zero, will become, by this va- 
riation, (a^x-^b^y+c^z, 8zc.y, A similar consequence 
wiU follow for E'^, E"^, &c. Hence it is evident, that the sum 
of the squares of the errors will have for its variation a quantity 
of the second order, with relation to ^y, &c., — a principle 
which perfectly accords with the nature of the minimum. 
If, after having determined all the unknown quantities a?, y^ 
&c., we substitute their values in the proposed equations, we shall 
obtain the different errors E, E' E", &c. belonging to the sys- 
tem, and which cannot be reduced, without augmenting the sum 
of their squares. If, among these errors, some are judged to be 
too considerable to be admitted, the equations which have pro- 
duced them should be rejected, as resulting from experiments of 
too defective a kind ; and we shall determine the value of the 
unknown quantities, by means of the remaining equations, which 
will then present errors of a much smaller kind. And, it is to 
be observed, that, though we reject the equations which have 
produced these errors, we shall not be obliged to retrace all the 
steps of the computation ; for, as the equations of minimum are 
ormed by the addition of the products belonging to each of the 
proposed equations, it will be only necessary to remove from 
that addition those products produced by the equations which 
have led to the errors. 
The rule by which we take the mean between the results of 
different observations, is but a very simple consequence of our 
general method, and which we shall term the Method of Mini- 
mum SaUARES. 
If experiment has given different values a\ a"\ &c. for a 
certain quantity a?, the sum of the squares of the errors will be 
(«' — xY 4 - — aY + — ^Y " 1 “ ’ 
