34 



Developing this last in powers of tp there results, as / «• = 0. 



•^ ^1.2 ^ 1.2.3.4.. ^ '^ 



or the even differential co-efficients must all vanish on the supposi- 

 tion ^=:t. These conditions are satisfied by supposing 



f ^ — A sin. ip + 5 sin. 2 ?i + &c 



But any function of this form vanishes in n readings. 



It is however to be feared, that even this degree of regularity 

 seldom obtains, from the unequal expansibility of the different 

 pieces of the instrument, which evidently cannot be reduced to any 

 law : still there must be an approximation to it, and therefore to 

 the correction of the error. 



5. It has become fashionable to talk of the flexure of instruments 

 since Bessel and Zach noticed its operation in their circles ; and 

 indeed it is a considerable temptation to admit its existence, that 

 astronomers of acknowledged character find discordances for which 

 they cannot otherwise account ; but it is too much like the occult 

 qualities of the Peripatetics to be freely admitted in these days. In 

 quadrants it certainly was often a cause of great inaccuracy, pro- 

 ducing permanent change of figure, but in circles it must occur in 

 a far less degree, and be corrected by opposite readings. In the 

 circles constructed by the German artists, where the telescope is 

 supported in the middle only, projecting far beyond the limb, it is 

 not surprising that its extremities should be unequally bent, notwith- 

 standing the counterpoises that are applied to the tube.* But in 

 those of this country, a single glancp at the framing is sufficient to 

 shew that the telescope cannot sensibly bend, much less the circle, 



♦ Zach once found a flexure in his telescope of three minutes, when the counterpoises wer^ 

 removed. 



