164 Mr. W. Galbraith's Remarks on the Experiments 



considered as a very close approximation to the truth. If this 

 is substituted in equation (8), it becomes 

 , _ 20919576 



y* 



(9) 



2 20919576+1856062635 2 

 Consequently it is now only necessary to determine y and z* 

 by means of equations (2) and (3) from the best observations 

 on the pendulum, to obtain e. 



In the Philosophical Transactions for 1819, Captain Kater 

 gives the following series of experiments on the pendulum : 



If in formula (1) we substitute the values sin 2 X, it will give 

 the length of the pendulum at each place where the experi- 

 ments have been made, when 3/ and 2 are known. In de- 

 signating by e iy e 2) e 3i &c. the differences between the lengths 

 observed and calculated by the formula, we shall obtain the 

 following equations of condition : 



1. 39-17146 -2—0-76136503/=^ 



2. 39-16159— 2— 0-7142003 y—e % 



3. 39-15554— z-0-6869483 y— e 3 



4. 39-14600— z—0-64555O45/= e 4 



5. 39-14250— 2-0-62460303/= e 5 



6. 39-13929-2-0-61279663/= e 6 



7. 39-13614— 2-0-5975166 y= e 1 



Now, to determine the values of 3/ and 2 by the method of 

 minimum squares, that is, in such a manner that the sum of 

 the squares of the errors e lt e 2 , e 3 , &c. may be the smallest 

 possible, it is necessary to form the equations of minimum 



* If a mean of Kater's, Biot's, Sabine's, and Goldingham's measures be 

 taken, t =0*008638 — , a very simple expression, though in our deter- 



R 



mination we have preferred the values of y and z deduced immediately 

 from the observations. 



with 



