1842.] On Equations of Condition for a Quadrilateral. 35 



tities ; and also that making adjustments by the sinal equations by 

 means of Tables as in the final calculations to seven places of decimals, 

 the difference of a single unit in the last place of a logarithmic sine 

 corresponds at 45° to an angular difference of ~ » " or about five hun- 

 dredths of a second ; and at 60° to - » or about eight hundredths of a 

 second ; moreover, it is well known that in adding together several 

 logarithms each of which is only approximately true in the last place, 

 there can be no certainty that the sum will be true within one or two units 

 in the last place ; therefore the difference between the two methods 

 of adjustment, (if I do not greatly err in estimating it,) may be consi- 

 dered as of no practical importance, being beyond the reach of the 

 Tables. 



In regard to the exceedingly minute quantities which some of the 

 continental observers used to profess themselves able to determine by 

 means of the repeating circle, there is a very sensible remark by old 

 Troughton, in a paper of the Astronomical Society, the substance of 

 which I quote from memory, to the effect that whatever be the ability 

 of the observer, or the construction of his instrument, he never would 

 believe in the quantities deduced beyond such as were visible in the 

 telescope. In fact, so long as observations have error at all, dispose of 

 that error how we may, we cannot get rid of it so as to ensure cer- 

 tainty ; the only advantage which the arrangement ultimately adopted 

 can possess is, that of being a little better than a number of other 

 arrangements equally possible, each of which is only somewhat less 

 probable. 



N. B. — In the equations (B) (C) (p) (o) the characters £<? £%> £* 9 £4, 

 denote the excess on the respective triangles. 



SPHEROIDAL EXCESS. 



To find an expression for the Excess on a Spheroidal Triangle. By 

 Captain Shortrede, 1st Assistant Grand Trigonometrical Survey. 

 It has been usual to consider the excess on a spheroidal as not dif- 

 fering sensibly from that on a spherical triangle of the same area as 

 estimated by the mean radius of the earth, and this may generally be 

 considered sufficient, for in the largest triangle ever observed, the 



