VOL. XX. j PHILOSOPHICAL TRANSACTIONS. 313 



Dejnomtration. — Call the ordinate mn, z ; and the subtangent Ao or me, s ; 

 also to the axis ar let another curve hgq be constructed, whose equation is 

 2 sz =x-, where its ordinate gm is = :r ; then this will be the quadratrix of the 

 logarithmic curve, according to the principle of my method ; that is, its sub- 

 normal is equal to the respective ordinate of this : as will appear by a calcula- 

 tion according to that method. Therefore, by what I have elsewhere explained, 

 if GC be drawn perpendicular and equal to gm, and hd parallel to go, meeting 

 the lines gm, cm in b and d ; then there will be the trapezium gbdc = aonm. 



But GBDC = GMC — BMD = ^X^ — \ BM^ =z sZ — -^ HA^ ; but HA = a/ 2 AO' fVom 



the nature of the curve hgq, therefore gbdc ■=■ sz — ao' = ao X mn — ao = 

 ME X MN — ME = ME X EN. Therefore a) SO AO MM = me X en. a.E.D. 



On the Gall Bee. Bij Mr. Benj. Allen. Isl° 245, p. 375. 



In some Aleppo galls which the insects* had not eaten their way out of, I 

 found a sort of bee, resembling the same sort of our wild bees which earth : 

 they have long wings ; the belly thick, and deep ; and on the back near the 

 commissure to the body, it is of a greenish black, the rest reddish, near a cinna- 

 mon colour. These galls were very gummy, some of them had a stem, and 

 the cavity round them was so extremely so, that not the least entrance to it 

 appeared, though the bee was beginning to make its way out ; which may seem 

 to show, that the atmospherical air is not necessary to the essence of life, be- 

 fore the organs of the body are employed ; but that it is maintained by a more 

 subtile air, that pervades the more minute pores, as it is convcA'ed to fish through 

 the water, &c. But this is not the only insect that I have found in galls, for in 

 the greyer sort, not so rich in gum, I have found a small ichneumon of a bright 

 green. 



On the ScarabcEus Galealus Pulsator, or the Death Watch.\ By Mr. Benjamin 



Allen. N° 245, p. 376. 



The second animal I observed is that which makes a noise resembling a watch; 

 it lived 4 days with me, beating exactly. I took one some years since, which 

 I then traced by the noise, as I did this, and they were both the same : from its 

 noise it has obtained the name of a death watch. I found it in a copper ; it re- 

 sembled dry dirt in colour. I found another some years before on a rotten 



* The insects which cause the galls, or globular swellings on the leaves of the oak, belong to the 

 Linnsean genera of cynips and ichneumon. 



f P^iniw /(jfirfiMM, see Naturalist's Miscellany, vol.3. Ptinus Pulsator, Linn. Gmel. Syst. Nat- 

 p. l605. Dermestu tesselatus, Fabr. Syst. Ent. p. 56. 



