150 PHILOSOPHICAL TRANSACTIONS. [aNNO 1726. 



square of the fourth term, must always be greater tliau the product of the 

 terms adjacent to them. 



Cor. Multiply either the square of the second term, or the square of the 

 fourth term, of a biquadratic equation, by -S-, and if the product does not 

 exceed the product of the adjacent terms, some of the roots of that equation 

 must be impossible. 



Prop. 5. In an equation of any dimension expressed by m, the coefficients 

 of the second, third, last, last but one, and last but two terms, being re- 

 spectively A, B, E, D, c, if the roots of the equation are all real, then shall 

 m — 1 X A- always be greater than 2mB, and m — 1 X d' greater than 



27WCE. 



1. For supposin g the roots to be a, h, c, d, e, &c. then by Lemma 4, shall 

 m — IX «' + m — 1 X i' -f OT — 1 X c^ &c. be greater than 2ab -\- 'lac 

 + lad, &c. and adding 2m — 2 X ah + 2m — 2 X ac -{- ^tn — 2 X ad, &c. 

 to both, the sum m — 1 X a^ + 2;« — 2 X ab -\- 7ii — I X Ir + &c. { = 'm'~l 

 X a -^-b -\- c,&c.") must be greater than 2mab + 2mac + 2mad, &c. that is, 



m — 1 X A- must be greater than 2mB. 



2. In general, it follows from this demonstration, that the square of the 

 sum of any quantities whose number is m, multiplied by m— 1, must be 

 greater than the sum of all the products can be made by multiplying any two 

 of them, multiplied by 2m. But it is easy to see, from the genesis of equations, 

 that CE is the sum of the products that can be made by multiplying any two 

 of the terms whose sum is d: from which it follows, that m — 1 X d'^ must be 

 always greater than 27«ce. 



Observations on the Dissection of an Ostrich. Bij Mr. George Warren, 

 Surgeon in Cambridge. N° 394, p. 113. 



Dr. Brown has so well described the parts of the ostrich he dissected 

 (Philos. Collect. N° 5*) that Mr. W. thinks there is not much to be added. 

 But the Dr. affirms it has no epiglottis ; yet in this subject that cartilage was 

 plainly visible ; and indeed the rimula appeared too open not to require one. 

 The OS hyo'i'des is 3 inches long from the basis; the musculi directores asperas 

 arterias were very plain, large and strong; the ring composed of 3 cartilages at 

 the divarication of the aspera arteria very bold; the 2 glands on the carotid 

 arteries, as large as small eggs. There was nothing in the lungs or heart, but 

 what it has in common with other birds. The 2 stomachs, viz. the crop and 

 gizzard, were filled witli half-digested grass, in which were some nails, some 



* Vol. il. p. J3i, of these Abridgmeiils. 



