1878.] Solution of Simultaneous Linear Equations. 113 



removed, and it is advisable to make the tensions each equal to half 

 the weight of one of the pulleys with its adjustable frame.) The 

 machine is now ready for use. To use it, pull the cords simultaneously 

 or successively till lengths equal fco e x , e 2 , . . . e n are passed through 

 the rings E 1? E 2 , . . . E„, respectively. 



The pulls required to do this may be positive or negative ; in 

 practice, they will be infinitesimal, downward or upward pressures 

 applied by hand to the stretching weights which (§) remain perma- 

 nently hanging on the cords. 



Observe the angles through which the bodies B x , B 2 , . . . B» are 

 turned by this given movement of the cords. These angles are the 

 required values of the unknown a?i, %%, . . . x n , satisfying the simul- 

 taneous equations (I) 



The actual construction of a practically useful machine for calculat- 

 ing as many as eight or ten or more of unknowns from the same 

 number of linear equations does not promise to be either difficult or 

 over-elaborate. A fair approximation being found by a first applica- 

 tion of the machine, a very moderate amount of straightforward 

 arithmetical w T ork (aided very advantageously by Crelle's multiplica- 

 tion tables) suffices to calculate the residual errors, and allow the 

 machines (with the setting of the pulleys unchanged) to be re-applied 

 to calculate the corrections (which may be treated decimally, for con- 

 venience) : thus, 100 times the amount of the correction on each of 

 the original unknowns, to be made the new unknowns, if the magni- 

 tudes thus falling to be dealt with are convenient for the machine. 

 There is, of course, no limit to the accuracy thus obtainable by suc- 

 cessive approximations. The exceeding easiness of each application of 

 the machine promises well for its real usefulness, whether for cases in 

 which a single application suffices, or for others in which the requisite 

 accuracy is reached after two, three, or more of successive approxima- 

 tions. 



December 12, 1878. 



W. SPOTTISWOODE, M.A., D.C.L., President, in the Chair. 



Dr. Philipp Hermann Sprengel w T as admitted into the Society. 



The Presents received were laid on the table, and thanks ordered for 

 them. 



The following Papers were read : — 



