ON THE ELIMINATION OF ARBITRARY FUNCTIONS. 233 



heights ovcrhauging the Dead Sea with a view to Nebo. The modern Jfebbeh 

 affords exactly the view described iu Deuteronomy, and I can find no other to 

 rival it. The city of Nebbeli is lo-\ver down on a spur of the range, and with 

 remains more perfect than ordinary. The whole country is here densely 

 crowded wuth ruins ; but the names do not indicate their ancient equivalents 

 — Maslubiych, Kuseir, Et Tein, &c. 



From Nebbeh w-e worked to Ayun, Mossa, Heshban, &c., which have been 

 visited by many others. We made some sojourn in the Seisaban, and iden- 

 tified Kamah, I3eth Jesimoth, and other scriptural sites, and thence worked 

 down the shore of the Dead Sea towards Callirrhoe. We ascertained that 

 the Seisaban, the ancient plains of Shittim, is of very much greater extent 

 than the maps represent. The fertile Ghor extends from the Beit Nemeirah, 

 or iipper fords, to within 3 miles of the mouth of the Callirrhoe, and is well 

 watered throughout ; but in ancient warfare this region could never be 

 defended, and the ruins are unimportant, though there is not a single mound 

 without the stones which tell of some fort of the olden time. 



We trust w"c have by our expedition carried out the intentions of the 

 British Association. We have carefully mapped . the whole country north of 

 the Arnon, every previous map of which we found to be a mere work of the 

 imagination. We have left no ruin in that tract unexplored ; and though we 

 have brought home no Moabite stones, we never dreamt we should be able 

 to do so. The grant was iov georirajjlikal exploration, and that we have endea- 

 voure'd cbnscientio'usl'y tb chrr)^ out, and have brought to light some twenty 

 ancient cities hitherto un visited and unknown, and others known only by name. 

 The zeal of my companions enabled me to exhibit about 100 photographs. 



Sur I' elimination des Fonctions Arbitraires. 

 By Ch. HermitEj Corr. Member of the Mathematical Society, London. 



[A communication ordered by the General Committee to be printed in extenso.'] 



C'est la definition geometrique d'une famille de surfaces par uu certain mode 

 do generation qui a conduit ix definir analytiquement une fonction z de x et 

 y par le systeme de deux equations 



^(^, y, z, n, A, B, . . . L)=0, 1 



,l.(.r,2/,r, a,A,B, ...L)=0,J '^ ^ 



ou entrent un parametre variable a et un nombre quelconque n de fonctions 

 arbitraires de a, representees par A, B, . . . L. Obtenir une equation aux 

 diflcrenccs partielles, a laquelle satisfait la fonction z quels que soicnt a et 

 ces Ji fonctions, sera la question traitce dans cette note par une methode 

 nouvelle. 



J 'observe en premier lieu que les relations donne'es permettcnt de conside'rer 

 X ct y comme dcs fonctions de z, dont les derivees successives, 



, dx ,, cl-x ,„ (Px 



a; =_, X = -— , .r = . . . 



c(z uz' dz^ 



, dii „ cVy „, d'l/ 



