86 A New Method of Resolving certain Equations, 



length beneath a clump of bushes, along the course of the 

 fluid; while my strength was insufScient to make it pene- 

 trate at all in any other direction. And along the whole 

 fifty feet, the evidence of its having passed, was indisputa- 

 ble. How the fluid passed through the thirty feet from the 

 tree to the wall, may perhaps not be thought quite so cer- 

 tain ; as it left no signs of its passage above ground, and no 

 indubitable ones could be discovered below by thrusting 

 down a staff. But for myself, I cannot doubt the first part 

 of its course was similar to the latter part ; but passing be- 

 low a thick and strong turf, and perhaps a little deeper, its 

 course could not be so easily traced. If the fluid did not 

 pass under ground the first part of its course it must 

 have come out of the ground a few feet from the tree, 

 leaped thirty-feet through the. air to the wall, and without 

 leaving any trace of its influence on the post and rails, or 

 displacing the small stones which composed the wall,* sunk 

 quietly down through the wall to its foundations, and there 

 gone off as above described at right angles to the wall, in 

 the direction of a line from this spot to the tree. 1 can- 

 not doubt that it passed the whole way from the tree undej? 

 ground. 



Art. XV.— -A new method of resolving Equations of the 

 third and fourth degree. By Alexander. C, Twining. 



To resolve a general equation of the third degree. 



Let the given equation be, 



x^-\-3ax^-\-3bx-\-c=:o, andput z-\-r=ix -, 



Then we have, 



z'^-{-3r-\-a.z''-\-3r^-\-^ar-fb.z+r^+3ar^-{-3br-{'C~o, 

 Assume, (r+a)(r^-h 3ar^-^3b r-l-c) = {r^+Qar-\-by ; 



Expanding; r'* -\-4ar^ -jr 3b-j-3a^ .r^ -\-c+3ab,r-}'ac=^ 

 rM-4ar3-}-264-4a='.r2+ 4ab.r-\- 6%- 



And uniting: b -a^.r^-{-c^ab.r+ac~-b^=o. (A) 



^ One of the portions of the fluid after it forked, fell into a heap ofsimr- 

 !ai- stones, and threw them about very much. 



