VOL. LXIX.] PHILOSOPHICAL TRANSACTIONS. 495 



Mr. Euler gave the following resolution, x — ^7r+ 'H/^ -\- l/<r -\- l/r -\- &c. 

 where tt, f, o-, t, &c. denote the roots of an equation of « — 1 dimensions v"' ' 

 — pv"-^ -\- qv''~i — &c. = O. It is evident, that in this case the equation 

 whose root is x will have w""' dimensions; for let the roots of the equation z" — 

 1 = O be denoted by a,, |3, y, i, &c. then will the quantity ^tt have the n fol- 

 lowing values x^/TT, (3^7r, y^Tr, &c. and the same may be affirmed of the quan- 

 tities ^f, ^(T, ^T, &c. and consequently the quantity v'^ + V' ? ^''' '^'"^^^ n X n 

 different values; and in the same manner the root x = y'7r-|- ^^ + ^<r-{- 

 f/T + may be proved to contain n X n X n X n X &c. := ?i"-' roots; and con- 

 sequently in this resolution, in equations of superior dimensions, the number of 

 independent coefficients (w — l) will be very few in proportion to the number of 

 dimensions n""^, or (if we respect its formula) n"~'^ of the resulting equation. 



Let n = 3 ; then the equation resulting will rise to an equation of Q dimen- 

 sions, which has the formula of a cubic; for \etx=4^Tr-\-4^^ = a one root, 

 then will ni-i^" a & — — ~ a be 2 other of the Q roots, and conse- 



2 ■* 



quently the roots will be a^ — a^ X x^ — b^ X x^ — c^ = 0, which has the for- 

 mula of a cubic; and in general the above-mentioned equation of n""' dimen- 

 sions will, for the same reason, have the formula of an equation of n"-^. 

 dimensions. 



Let the resolution be:r= ^7r+ ^f+ ^(r + ^-^ + &c- where tt, f, o-, t, 

 &c. denote the roots of an equation x"- • — px"-'' -\- qx^—^ — &c. = O of (n— l ) 

 dimensions; then will the resulting equation free from radicals, whose root is x, 

 rise to 2°~' dimensions; but as every affirmative has a negative root equal to it, 

 it will have the formula of an equation of 2"~* dimensions. 



Let the resolution be of this formula x = '■+'"v'« + "•+'"■/ i3 '+"'\^y -{- '+'"/J 

 -i- &c. if x, (3, y, S, &c. be considered as the r power of the roots of an equation 

 of s dimensions, then will the resulting equation, of which the resolution is 

 given, rise only to an equation of the formula of m' ' dimensions. In the year 

 1762 I published some reasons, for which this method could not extend to the 

 general resolution of algebraical equations. 



XL Observations on the Total (with Duration) and Annular Eclipse of the Sun, 

 taken June 24, 1778, on Board the Admirars Ship of the Fleet of New 

 Spain, in the Passage from the Azores totvards Cape St. Vincent's. By Don 

 Antonio Vlloa, F. R. S., Commander. From the French, p. 105. 



A very favourable, though long, passage gave Don UUoa the opportunity of 

 observing at sea the eclipse of the sun, which was accompanied by a phe- 

 nomenon observed by few astronomers, viz. the luminous annulus which 

 surrounds the disc of the moon in such an eclipse. The motion of the ship 

 prevented observing the beginning of the eclipse, by reason of the difficulty 



